Paleoceanography - 3-Box Model

1 INTRODUCTION TO OCEAN CARBON SYSTEM BOX MODELS: THE TOGGWEILER-SARMIENTO 3-BOX MODEL A 3-box model is the simplest model that illustrates most of the principles involved in construction of ocean carbon system chemical models. The model presented below is based most closely on that by Toggweiler and Sarmiento (1985), but is not exactly the same (changes have been made to simplify the presentation while retaining most of the functionality). Toggweiler and Sarmiento didn't compute their model in matrix form - they worked out the algebraic equations. However, it is much simpler to set up a model in this matrix form once you get used to it - it took me a couple of days to set up most of this model, half of which was spent in getting the units correct! Getting the units consistent at the outset is worth the effort; it will save a lot of debugging later. After the basic model is set up, two scenarios are illustrated. The first scenario is a "pre­ industrial atmosphere" scenario where pCO2 = 280 ppmV and the ocean carbon system is adjusted to be consistent with observations on the modern ocean (corrected for fossil fuel CO2 uptake, etc.). The second scenario is a hypothetical "glacial" scenario where some changes between glacial and interglacial times are driven by observation (e.g. atmospheric CO2 was ~200 ppmV) and others were made as assumptions. The salinity of the ocean increased due to the withdrawal of water into continental glaciers (and the average concentrations of phosphorus, alkalinity and dissolved carbon dioxide increased proportionately, while the mass of the ocean decreased). A transfer of carbon to the ocean from oxidized continental carbon was assumed (estimated from mean oceanic δ13C) and this extra CO2 was "neutralized" by stoichiometric dissolution of CaCO3. Warm surface T decreased from 21.5°C to 20°C. Cold surface T decreases from 2.5°C to 2.0°C (somewhere between the realm of observation and hypothesis). 2 SALINITY: Although it is of only minor relevance to the carbon system, the salinity distribution provides the easiest illustration of the model equations. The processes include mixing between boxes, and a transfer of water from low latitudes to high latitudes through the atmosphere. Underlying this salt balance is the water balance, where water fluxes in and out of each box must balance. F Q13 Q12 1 Q21 Q23 Q31 2 Q32 3 Consider the steady-state salinity mass balance for the high latitude box 2: +Q12 S1 - Q21 S2 - Q23 S2 + Q32 S3 = 0 which can be slightly rewritten to: +Q12 S1 + (-Q21 - Q23) S2 + Q32 S3 = 0 which can be expressed as a vector equation: | +Q12 (-Q21 - Q23) +Q32 | |S1| |S2| = 0 |S3| Writing the equation for box 2 and the overall mass balance equation in this form, and collecting them together into a matrix equation: the matrix formula describing this system of simultaneous equations is: A x = b | -Q12-Q13 +Q21 +Q31 | | S1 | | 0 | | +Q12 -Q21-Q23 +Q32 | | S2 | | 0 | +V2 +V3 | | S3 | | Stot | | +V1 The first two equations describe the mass balances for boxes 1 and 2, and the third equation is the salt mass balance for the system as a whole (an equation for box 3 is not necessary since it is a linear combination of the equations for boxes 1 and 2). Note that the atmosphere is assumed to have no salt and as no significant reservoir of water. 6 3 discussion of units: water fluxes in the ocean are typically expressed as "Sverdrups", 10 m /sec. However, concentrations of chemicals are typically expressed in units of moles/kg. To keep the units consistent, we multiply water fluxes by the density of seawater, 1027 kg/m . This conversion factor is an approximation, because the density of seawater depends slightly on temperature, salinity, and pressure, which must be taken into account for very precise applications. Also note that the average salinity of the ocean is tied to the mass of the ocean - changes in mean salinity are created by withdrawing water from the ocean into massive continental glaciers. Any model which assumes a change in mean ocean salinity has 3 to include an inversely proportional change in the mass of the ocean, which will impact the concentrations of other dissolved chemicals. Another point to note is that our choice of the water fluxes and system volume as the master variables in "A" is a contextual choice; they could have been included as variables. In our case, the fluxes and volumes are chosen as master variables because they apply to all constituents - hence, once "A" is set up, it applies to all of the subsystems. 6 Carbon Dioxide: | -Q12-Q13 +Q21 +Q31 | | C1 | | fc1Fp1 + Fca1 | | +Q12 -Q21-Q23 +Q32 | | C2 | | fc2Fp2 + Fca2 | | +V1 +V2 +V3 | C3 | | Ctot - PCO2 Matm | A | x = b The CO2 model is similar to the alkalinity model, except that CO2 is transported through the atmosphere and the atmosphere is a significant reservoir. The introduction of the atmosphere as a reservoir would best be handled by adding a fourth box (this is how it is implemented in the Excel spreadsheet; it makes the extension to C13 and C14 much easier). However, to stick to our simple 3-box formulation for this tutorial, we handle this issue simply by removing the carbon dioxide content of the atmosphere from the total contained in the ocean system (bottom term of b). The particulate carbon flux is affected both by calcium carbonate formation/dissolution and organic matter formation/degradation. The fluxes of carbon dioxide between the ocean and atmosphere (Fca1 and Fca2) are shown as both going into the atmosphere for the sake of consistent notation; at steady-state these will be equal to each other in magnitude but of opposite signs. Because pCO2 and hence gas exchange fluxes are non-linear functions of Alk and CO2, it is necessary to solve this system by iteration; we guess atmospheric pCO2 and air-sea fluxes, solve the linear system, and then modify the guesses iteratively until the system converges. Note also that because we are comparing fluxes calculated per unit of volume transfer (fluxes between ocean boxes) and fluxes calculated per unit of surface area (gas exchange as a function of the partial pressure difference between the ocean and atmosphere), it is especially important to make the units consistent. 8 Carbon 14: -Qз1 -Qз2 - l+о1з л, о 14сз I l -f14c1Fp1- f14c2Fp2 о А х = ь The 14С model is almost the same as the 1зс model. Two changes аге very minor: (1) converting the equilibrium and gas exchange constants to the 14С values from the 1зс values, and (2) adding а radioactive decay term to each Ьох (-л.С). The Ьiggest change is in the 4th 14 equation for mass ba la nce. Beca use с is radioactive, it is not conserved and its total inventory оп earth сап change as cosmic гау production increa ses. This factor could Ье handled in different wa ys; in this version we simply require that the tota l 14С flux across the air/sea boundary is equal to the amount produced Ьу cosmic rays (less the small amount which decayed while in the atmosphere ).