Box Models – Problem Set 1
Due Feb. 26, 2013 - you can consult others or work together if needed, key thing is to learn!
This is a simple introduction to box modeling of ocean chemistry based on the Toggweiler and Sarmiento (1985) box model. These exercises will get you acquainted with the concept and design of simple box models, and the idea of geochemical signatures of the “Biologic Pump” as illustrated below:
The idea is to develop an understanding of how conservative and non-conservative elements are handled in a box model. We will soon use Ed Boyle's Excel-version of the Toggweiler and Sarmiento (1985) model to explore the sensitivity of past oceans (and the atmosphere) to prescribed changes in circulation, productivity, or bulk chemistry as a way to interpret the geologic record of Cenozoic climate changes in a more quantitative way.
Ed has very kindly shared the model with us and I wanted you to know how the solver works. In short, he's replaced a bunch of linear equations for each constituent (Salinity, PO4, Alkalinity, d13C, CO2, CO3= etc) with matrix algebra and it makes the whole thing a lot … cleaner.
Below is a depiction of three box model we're considering. Box 1 is the "surface tropical ocean" and represents 2% of the ocean volume, Box 2 is the "surface polar ocean" and represents 1% of the ocean volume; the remainder of the ocean (97%) is of course the deep-sea below the thermocline.
Total mass of the ocean = 1.292E+21 kg
Average salinity of the ocean = 34.7 psu
Mass balances for each box are determined using the following criteria: fluxes into (out of) a box are positive (negative). Since we're considering a steady state, conservative element here the net fluxes into and out any box are equal to zero.
Qmn = water fluxes from box m to box n, in kg/year
S_ n = salinity in box n
Water fluxes (Qmn) in m3/year are in Sverdrups (10^6 m3/s). Water mass flux (Vmn) can be calculated based on observed flow (in Sverdrups, 10^6 m3/s), times seconds/year, times density of sea water (1027 kg/m3).
Q12 = 29 Sv "poleward gyre flow" (10^6*sec_yr*kg_m3) V = 9.38165E+17 kg/year
Q13 = 1 Sv "subtropical downwelling" (10^6*sec_yr*kg_m3) V = 3.235E+16 kg/year
Q31 = 20 Sv "equatorial upwelling" (10^6*sec_yr*kg_m3) V = 6.4701E+17 kg/year
Q21 = 10 Sv "equatorward gyre flow" (10^6*sec_yr*kg_m3) V= 3.23505E+17 kg/year
Q23 = 72 Sv "deep water formation" (10^6*sec_yr*kg_m3) V= 2.329E+18 kg/year
Q32 = 53 Sv "subpolar upwelling" (10^6*sec_yr*kg_m3) V= 1.71458E+18 kg/year
For simplicity, you can calculate salinity mass fluxes as S_n * Qmn (i.e. salinity * water flux in kg/year; no need compute actual salt mass)
1a. Write the salinity mass balance equation for Box 1.
1b. Write the salinity mass balance equation for Box 2.
1c. Write the salinity mass balance equation for the total ocean, where the sum of all boxes (Vn * Sn, where Vn is mass of water in box n) is equal to the global ocean salinity reservoir (S_tot).
1d. Note that all three equations have S_1, S_2, and S_3 as variables. Now, write these three equations as a single matrix equation: A x = b, where:
A = 3 x 3 matrix of fluxes (from equations a, b, c above)
x = 1 x 3 vector of salinity values for each box, S_1, S_2, S_3
b = 1 x 3 vector of [0, 0, S_tot)
1e. Now, use the attached Excel spreadsheet to solve for x (the vector of salinities in each box). Populate the yellow cells with the numbers calculated in questions 1a-c and 2a. The Excel function for inverting a matrix is MINVERSE(x) and the function for multiplying matrices is MMULT(x,y).
Use Excel to solve x = A-1 b in the red area. I wanted you to use Excel rather than Matlab because the box model we'll be using is also in Excel.
1f. What are the calculated salinities in boxes 1, 2, 3 ? (unsurprising answer)
2. Non-conservative tracers (such as phosphate, PO4). The purpose of part 1 was to prep you for calculating between-box fluxes of non-conservative components such as PO4, d13C, TCO2, etc. In the modern ocean, the sinking and remineralization of organic matter from the surface transfers nutrients and carbon to the deep sea (biologic pump). Rather than model the biological system explicitly, we simply take the modern phosphorus distributions in the ocean to constrain the implied particulate P sinking flux between boxes 1 and 3 (Fp1) and between boxes 2 and 3 Fp2 (note green labels on xls spreadsheet.
Instead of the conservative behavior we considered in question 1 where the sum of the salinity fluxes into/out of a given box were defined as zero, now the net PO4 flux of a box can sum up to a positive value equivalent to either Fp1 or Fp2, the particulate PO4 export to deep-ocean box 3.
2a) Write the matrix equation (in notation form e.g. V13 * P1 + …) for the PO4 mass balance using the Ax=b approach you used in question 1d (first row is box 1 mass balance, 2nd row is box 2 mass balance, and third row is total ocean PO4 inventory mass balance. The average PO4 concentrations for Boxes 1, 2, and 3 are 0.20, 1.42, and 2.15 µmol/kg, respectively.
2b) Okay, now solve for b! What are the values for Fp1, Fp2, and Ptot? (F units are µmol/yr, and Ptot units are µmol).
2c) Which box has the larger particulate PO4 flux to Box 3? How would you interpret this oceanographically, (i.e. why would the flux be larger here?).
2d) Now let's try a thought-experiment. Let's assume that we now set the average PO4 concentrations for Boxes 1, 2, and 3 are 0.20, 0.20, and 2.16 µmol/kg, respectively. What happens to the particulate fluxes? How would you interpret these changed PO4 concentrations in terms of ocean productivity?
2e) For the results in 2c (today's ocean) the calculated atmospheric CO2 levels using a similar model with full ocean C-chemistry (the one you'll use soon) were similar to today, about 280 ppm. For the results calculated in question 2d the atmospheric CO2 levels were a lot lower, near 175 ppm. Why do you think this was?