Examples and Applications

The Global Scenario

The Revelle Buffer Factor and Alkalinity

Increasing Anthropogenic CO2

Contributions to CO2 Absorption by Fresh Waters


The Global Scenario

Transferring all these simulations for seawater equilibria to the whole ensemble of the oceans is far from straightforward. A complex pattern emerges as temperature, salinity and DIC vary from arctic waters to tropical ones. By way of example, DIC distribution on the ocean’s surface is depicted in Fig.1.










Fig 1. Present day surface DIC (from wikipedia/commons/d/df/WOA05_GLODAP _pd_DIC_AYool.png By Plumbago. Own work CCBY-SA 3.0)

As in the atmosphere, ocean streams (like the famous Gulf Stream) move huge masses of waters, which eventually sink (a fact known as vertical mixing). This vertical mixing in the ocean is driven by buoyancy. The two factors that determine buoyancy in the ocean are temperature and salinity. The sinking of waters from the surface to the ocean depths takes place in Polar Regions where the surface water is cold and salty, and hence heavier. This, in turn, determines a movement of tropical waters toward Polar Regions, which causes evaporation and an increase in salinity, thereby closing the cycle. In other regions, the ocean surface is warmer than the water underneath, so that any vertical mixing is suppressed. Some vertical mixing still takes place near the surface due to wind stress, resulting in an oceanic mixed layer extending up to around 100 m in depth and slowly exchanging with the deeper ocean. Dynamic variation (as discussed in Section 6.3) results in gaseous exchange with the air in the mixed layer (volume = 3.6·1016 m3) over a time scale of around 1 year. Equilibration of the whole ocean, for example in response to a change in atmospheric CO2, has a much longer time scale of around 120 years or more.

Therefore, in applying the algorithm to the oceans as a whole many precautions must be taken, always bearing in mind that, while CO2 concentration is practically uniform, there are marked temperature, salinity and DIC variations across the globe. From my own experience, one can divide the simulation task into two steps: first, by assuming a fixed value for some parameters, for example the average present-day DIC taken from fig. 1 as 2.06.; second, by finding out how the fixed parameter(s) change according to latitude or other local geographical parameters. An example will help clarify the procedure.

Fig. 2 Difference between present-day average DIC (green line) and equilibrium (saturation) DIC with 410 ppm, standard seawater composition.

As we know, there is an ocean uptake of CO2 in colder areas (high latitudes) and an ocean outgassing in warmer ones (tropics). The red curve in Fig. 8.2 corresponds to DIC saturation, as calculated with SeaWaterCalc for standard seawater composition, in equilibrium with 410 ppm CO2 (no precipitation of calcite) as a function of temperature from -2 to 30°C. If we compare the red curve with the horizontal line (green) corresponding to present-day average DIC (estimated at 2.06 mmol/kg), we can see how, for T<17°C, there can be an uptake of CO2 and an outgassing for T>17°C. We can improve our simulation by accounting for the variation of DIC around the globe (see Fig.8.1). Comparing the picture in Fig 1 with a temperature map of the ocean surface, one can assume that temperature and DIC have a linear relationship, and moreover (as first approximation) that at 0°C, DIC equals 2.20, decreasing then (linearly) to 1.93 at 30°C.

On this basis, the following linear relationship could be derived:

DIC = 2.20 – 0.009*T(°C)

When using this DIC dependence from temperature, one can obtain a more realistic picture of the CO2 flux, as shown in Fig. 3

Fig. 3 DIC Saturation (red) and present-day DIC for variable temperatures.

An evaluation of the DIC disequilibrium associated with the plot, ( ΔDIC = DICsat – DIC) gives results from 0.06(0°C) to -0.05(30°C), perfectly comparable with the values given in fig 6.15(b) page 147 of Williams 2011, which range from 0.06 to -0.06 for the ocean surface overall across the globe inferred from 'an independent global climatology'.

The key point is to transform this useful information on ocean DIC disequilibrium into CO2 in- or out-flux. This can be done using the 'famous' Takahashi plot with relative data, as diffused in literature and numerous books since 2002. It reports the air-sea CO2 flux based on a compilation of ocean surface observations from Takahashi et al. (2002). According to personal interpolation of the data from SeaWaterCalc and literature, the approximate correspondence between ΔDIC and the consequent annual mean CO2 flux is:

CO2 flux (mol m-2 y-1) ≈ 100* ΔDIC (mmol/kg-solution)

This approximate equation enables us however to estimate air-sea CO2 flux on the basis of relatively simple simulations and algorithms.

The empirical expression of DIC on the basis of temperature alone (thus disregarding the contribution of salinity), also allows us to calculate from DIC the concentration of CO2 in atmosphere at equilibrium at a given temperature. With a reference value of 410 ppm, if the equilibrium concentration has a lower value (see Fig. 4), as happens up to 23 °C, then seawaters are prone to absorb CO2. On the contrary, they will desorb it at temperatures over 23°C.

Fig. 4 In- and Out-flux of CO
2 as deduced by equilibrium CO2 calculation (red line) as a function of T and DIC.


The Revelle Buffer Factor and Alkalinity

The relationship between fractional changes in pCO2 and in DIC is formally expressed in terms of the Revelle buffer factor B, which is defined as the ratio between the relative changes in pCO2 and DIC (Bolin-Erickson 1959),

 B = (δpCO2/pCO2) (δDIC/DIC)

In other words, the Revelle buffer factor is a measure of the relationship between changes in pCO2 and consequent variation of DIC in seawater. Being the ratio between two pure numbers, it is a pure number in its own right. In today's ocean surface, B varies from 6 in the warmest tropical waters (30°C) to 16 in high latitudes or arctic waters (-2°C). As a consequence of its relevance to the scale of DIC increase compared to the increase of anthropogenic CO2, many attempts have been made to simulate or evaluate it using physico-chemical seawater characteristics.

Using the iterative approach and the routines described here, with some minor modification (not listed), anybody can calculate this 'B' factor. There is no need to introduce cumbersome and confusing-looking accessory concepts like alkalinity or carbonate alkalinity, so often viewed as necessary by many textbooks.

If the reader wishes to change the chemical composition of the seawater, this can be easily achieved in the input text file (SeaWaterCalc.txt) for each element or compound. Alkalinity is often cited and used because it is an experimentally measured parameter with a simple HCl titration of seawater but, thanks to the available algorithms, it can also be deduced from composition. In particular, Total Alkalinity (TA) is defined as the concentration of all the bases that can accept H+ when a titration is made with HCl to the carbonic end point, when all carbonate species are transformed into H2CO3 (around pH 3-4, titration with methyl orange). The following is an explicit version, taking into account the hydroxyl and oxonium ions:

TA = [HCO3-] + 2[CO3 - -] + [B(OH)4 -] + [H3SiO4 -] + [MgOH+] + [OH-] + [H+]

It can be simply calculated from the output file of the main program (untitled.txt)

Fig. 5 Temperature dependence of Revelle factor with (blue) no CaCO
3 precipitation and with (red) a precipitation factor of 0.01 (1%). Simulations with SeaWaterCalc for 410 ppm of CO2.

Increasing Anthropogenic CO2

The recent interest in the distribution of CO2 in the oceans is related to the need to understand how the increased amount of the emissions of this gas in the atmosphere will be buffered by carbonate equilibria in seawater. The partial pressures of CO2 in the atmosphere (pCO2) have been studied by a number of researchers. The classical measurements of pCO2 were first made by Keeling (Keeling-Worf 2004) at the Mauna Loa Observatory in Hawaii, back in 1958. More recent measurements have been made on the air trapped in ice cores up to 400 thousand years ago. These measurements clearly demonstrate that CO2 is increasing in the atmosphere because of the burning of fossil fuels. However, the final amounts in the atmosphere are only ≈52%, the oceanic sink accounting for ≈48% of the total fossil-fuel burning and cement-manufacturing (Sabinel 2004). In 2009 Khatiwala et al. derived, as they describe it, “an observationally based reconstruction of the spatially-resolved, time dependent history of anthropogenic carbon in the ocean over the industrial era (AD 1765 to AD 2008)” based on a suite of sampled ocean tracers (Fig. 6).

Fig. 6 Atmospheric CO
2 concentration and oceanic uptake rate for anthropogenic carbon (with shaded error envelope) plotted against time. Adapted from Khatiwala et al. 2009.

Interest has now evolved towards possible future trends. Using the SeaWaterCalc code, and interpolating the data of Mauna Loa with a second degree polynomial, to a certain point it is possible to foresee the future behaviour of some parameters, like in Fig 7, pH and pH with 1% precipitation of CaCO3

Fig. 7 Parabolic interpolation (black) of the Mauna Loa data (Keeling curve, red) from the start of measurements (1958) up to now. Based on the values of the interpolated black curve, the pH equilibrium values are calculated up to 2050 (blue curve without calcite precipitation, cyan line pptF = 0.01)

Contributions to CO2 Absorption by Fresh Waters

Rather surprisingly, in oceanic chemistry the CO2 absorption contribution by fresh waters is never considered. When meteoric water (rain water) flows or comes into contact with carbonate rocks (containing CaCO3), the dissolution of calcite (or aragonite) occurs spontaneously due to the high content of CO2 dissolved in rain water. In fact, this water has absorbed CO2 throughout the entire course of the formation and precipitation of its rain droplets.

As to a single reaction step, quite often the expression of the dissolution of calcite by meteoric waters is written as:

CaCO3 + CO2 + H2O Ca++ + 2 HCO3

It suggests that CO2 is consumed in a stoichiometric 1:1 manner when calcite is dissolved. However, being a reaction (..) only one part of a series of equilibria, and not a single step reaction, it is therefore necessary to consider all the equilibria (below) in the chemical system in an attempt to find a correct quantitative resolution.

1 – CO2 (gas) + H2O H2CO3 * (H2CO3 * is the sum of dissolved CO2 and H2CO3)

2 – H2CO3 H+ + HCO3

3 – HCO3 H+ + CO3– –

4 H2O H+ + OH

5 Ca++ +CO3– – CaCO3 (calcite)

As the empirical approximations of Keq described in Chapter 5 are no longer valid when salty water is substituted with fresh water, we should use another approach, referring ourselves to the appropriate Gibbs energy values from a literature database. Gibbs energy is also the chemical potential that is minimised when a system reaches equilibrium at constant pressure. As such, it is a convenient criterion of spontaneity for isobaric (constant pressure and variable volume) processes. Gibbs free energy, originally called available energy, was developed in the 1870s by the American mathematical physicist Willard Gibbs.
The change in Gibbs free energy, ΔG, in a reaction is a very useful parameter. It can be thought of as the maximum amount of work obtainable from a reaction at constant pressure (usually reactions occur at ambient pressure, 1 atm). For example, in the oxidation of glucose, the main energetic reaction in living cells, the change in Gibbs free energy is ΔG = 686 kcal = 2870 kJ. The change in Gibbs free energy associated with a chemical reaction is a useful indicator of whether the reaction will proceed spontaneously. Since the change in free energy is equal to the maximum useful work which can be produced by the reaction then a negative ΔG indicates that the reaction can happen spontaneously. Knowing the ΔG value of a reaction, the value of its equilibrium constant can be calculated. For a generic reaction like aA + bB = cC + dD it can be demonstrated (see also Section 3.3) that when equilibrium is reached:


Keq =


=  exp [-ΔG/(RT)]


The complete list of considered equilibria is already written above, their equilibrium constants are calculated from Gibbs energy values (data are taken mainly from NIST or other thermodynamic databases). Remember that K(eq) = exp(- ΔG/RT), R being the gas constants and T the absolute temperature. By using the thermodynamic Gibbs energy we can account for the temperature dependence of the equilibrium constants. The code needed to solve the system is in some way similar to SeaWaterCalc code in the iterative procedures, but different in the usage of numerical values of the equilibrium constants, as they are now calculated from Gibbs energy.

1- ΔG = -20302 – T*(-96.25) (Joule/mol/K)

2- ΔG = 7660 – T*(-96.2) (Joule/mol/K)

3- ΔG = 14850 – T*(-148.1) (Joule/mol/K)

4- ΔG = 55836 – T*(-80.66) (Joule/mol/K)

5- ΔG = -13050 – T*(-202.9) (Joule/mol/K)

From the above treatment of inorganic carbon chemistry in fresh water and the simultaneous resolution of temperature-dependent equilibria, interesting results are obtained. They are presented in graphic form in figures 8.8 and 8.9, for the sake of simplicity. One striking finding is that the molar amount of CO2 absorbed and calcium ions released in solution by calcite dissolution nearly coincide (green and blue curves). The increase in DIC is indicated by the red curve. It can reach values of up to 1.3 mmol/L at equilibrium, obviously when sufficient time is given. It is not so far from the common values for seawater (from 1.85 to 2.20), but it is likely that fresh waters are still a long way off from equilibrium when they join oceanic waters, thereby changing their composition completely.

Fig. 8 Trend of dissolution of calcite when ppm CO2 in the atmosphere grow from 280 (pre-industrial value) to 780. Temperature is 17°C in the simulation.

Fig. 9 Trend of dissolution of calcite when seawater temperatures increase from -2°C (arctic seas) to 30°C (tropical seas). CO2 ppm are 410 in the simulation.