3_box_model_Page_1.tif3_box_model_Page_2.tif3_box_model_Page_3.tif3_box_model_Page_4.tif3_box_model_Page_5.tif3_box_model_Page_6.tif3_box_model_Page_7.tifMIT OpenCourseWare http://ocw.mit.edu 12.740 PaleoceanographySpring 2008For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.INTRODUCTION TO OCEAN CARBON SYSTEM BOX MODELS: THE TOGGWEILER-SARMIENTO 3-BOX MODEL Spring, 2006 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 pC02 = 280 ppmV and the ocean carbon system is adjusted to be consistent with observations on the modern ocean (corrected for fossil fuel C02 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 C02 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 (estimatedfrom mean oceanic 613C) and this extra CO2 was "neutralized" by stoichiometric dissolution of CaC03. Warm surface T decreased from 21.5"C to 20°C. Cold surface T decreases from 25°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. Consider the steady-state salinity mass balance for the high latitude box 2: +Qq2 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: I +Q12 (-Q~I-Q23) +Q32 I IS1I IS21 = 0 IS31 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: 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 Iand 2). Note that the atmosphere is assumed to have no salt and as no significant reservoir of water. 6 3discussion of units: water fluxes in the ocean are typically expressed as "Sverdrups", 10 m Isec. However, concentrations of chemicals are typically expressed in units of molesikg. To keep the units consistent, we multiply water fluxes by the density of seawater, 1027 kglm . 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 has3 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.~,1and ~,2 are the fluxes of particulate biogenic phosphorus from boxes 1 and 2 into box 3, where they quantitatively decompose back into dissolved form ("regeneration"). The mass balance is: Advective Fluxes + Particle Fluxes = 0 (for each box.) The sign of the particle flux in the b matrix for boxes 1 & 2 is positive because it has been moved from the left side -of the equation -to the right; e.g., the P equation for box one is: +Q12 PI Q21 P2 Q23 P2 + Q32 P3 -Fpl = 0 The particle flux is the negative term (at the end of the leA hand side of the equation) switches sign when you move it to the right hand ("zero") side of the equation: +Q12 PI -Q21 P2 -Q23 P2 + Q32 P3 = Fpl How do we derive the biological particle fluxes? Rather than model the biological system explicitly, we simply take observations (or hypotheses) of the phosphorus distribution in the ocean to constrain PI, P2, and Pt,t. (which sets P3 from the mass balance). The equation is used to calculate Fpl and Fp2 (Ax=b). Low-latitude surface water P is depleted to low concentrations by efficient biological uptake (in other words, P is the model "limiting nutrient"), and high-latitude surface water P is taken as a master variable to be explored. Note that it cannot be varied completely arbitrarily; some values might imply an upward flux of particles! Also, Although the ocean phosphorus concentration is not mechanistically dependent on salinity, a withdrawal of water from the ocean (resulting in higher salinity) will increase the mean dissolved phosphorus concentration.ALKALINITY: The alkalinity model is similar to the phosphorus model, except that the alkalinities are not fixed by observation or hypothesis but instead calculated after fixing the "Redfield" ratio