3.3 Nutrients & Biogeochemical Cycles

Step 1: Average Elemental Composition of Marine Organic Matter

Redfield (1934) analyzed the elemental composition of marine plankton and dissolved nutrients in seawater. He found that the molar ratios of C, N, and P in marine organic matter are remarkably constant. We write the generalized formula for marine organic matter as:

$$(\text{CH}_2\text{O})_a (\text{NH}_3)_b (\text{H}_3\text{PO}_4)_c$$

Step 2: Determine Stoichiometric Coefficients from Observations

From thousands of measurements of dissolved nutrient drawdown during photosynthesis and regeneration during remineralization, the observed consumption ratios give:

$$\frac{\Delta[\text{NO}_3^-]}{\Delta[\text{PO}_4^{3-}]} \approx 16, \quad \frac{\Delta[\text{CO}_2]}{\Delta[\text{PO}_4^{3-}]} \approx 106$$

Step 3: Oxygen Demand from Complete Remineralization

To find the O₂ consumed during complete oxidation of organic matter, we balance the redox reaction. Oxidizing one mole of carbohydrate (CH₂O) requires one O₂. Oxidizing NH₃ to HNO₃ requires 2 O₂ per NH₃. The total oxygen demand is:

$$\Delta \text{O}_2 = 106 \times 1 + 16 \times 2 = 106 + 32 = 138 \text{ mol O}_2$$

Step 4: The Complete Redfield Equation

Combining all terms, the stoichiometrically balanced equation for photosynthesis (forward) and remineralization (reverse) is:

$$106\text{CO}_2 + 16\text{HNO}_3 + \text{H}_3\text{PO}_4 + 122\text{H}_2\text{O} \underset{\text{respiration}}{\overset{\text{photosynthesis}}{\rightleftharpoons}} (\text{CH}_2\text{O})_{106}(\text{NH}_3)_{16}\text{H}_3\text{PO}_4 + 138\text{O}_2$$

Step 5: Verify Water Balance

We verify by counting hydrogen and oxygen atoms on both sides. On the left: 122 H₂O provides 244 H atoms. CO₂ contributes 212 O atoms. On the right: 106 CH₂O contains 212 H and 106 O; NH₃ contributes 48 H; 138 O₂ provides 276 O. Balancing confirms the 122 H₂O coefficient and the C:N:P:O₂ = 106:16:1:138 ratio.

$$\boxed{\text{C}:\text{N}:\text{P}:\text{-O}_2 = 106:16:1:138}$$

Step 1: First-Order Remineralization Kinetics

Particulate organic matter (POM) sinking through the water column is decomposed by heterotrophic bacteria. The rate of organic matter decay follows first-order kinetics with respect to the organic matter concentration:

$$\frac{d[\text{POM}]}{dt} = -k_{\text{remin}} \cdot [\text{POM}]$$

Step 2: Solve the ODE for Exponential Decay

Separating variables and integrating from the initial concentration at export depth z₀:

$$\int_{[\text{POM}]_0}^{[\text{POM}]} \frac{d[\text{POM}]}{[\text{POM}]} = -k_{\text{remin}} \int_0^t dt \quad \Rightarrow \quad [\text{POM}](t) = [\text{POM}]_0 \cdot e^{-k_{\text{remin}} t}$$

Step 3: Convert Time to Depth Using Sinking Speed

For a particle sinking at velocity w, the time to reach depth z is t = (z - z₀)/w. Substituting yields the Martin curve (power-law flux attenuation):

$$F(z) = F(z_0) \cdot \left(\frac{z}{z_0}\right)^{-b}$$

where b ≈ 0.858 (Martin et al., 1987). This arises when the remineralization rate and sinking speed both vary with particle size and composition.

Step 4: Nutrient Release Rate from Redfield Coupling

Using the Redfield ratio, the rate of nutrient regeneration is coupled to carbon remineralization. For nitrogen and phosphorus regeneration:

$$\frac{d[\text{NO}_3^-]}{dt} = \frac{16}{106} \cdot k_{\text{remin}} \cdot [\text{POC}], \qquad \frac{d[\text{PO}_4^{3-}]}{dt} = \frac{1}{106} \cdot k_{\text{remin}} \cdot [\text{POC}]$$

Step 5: Apparent Oxygen Utilization (AOU) Connection

The oxygen consumed during remineralization (AOU) is related to nutrient regeneration through Redfield stoichiometry. This allows estimation of regenerated nutrients from oxygen measurements:

$$\text{AOU} = [\text{O}_2]_{\text{sat}} - [\text{O}_2]_{\text{obs}} = \frac{138}{16}\,\Delta[\text{NO}_3^-] = 138\,\Delta[\text{PO}_4^{3-}]$$