3.4 Ocean Acidification

Step 1: Thermodynamic Basis of Saturation

The dissolution of CaCO₃ is governed by the Gibbs free energy. At equilibrium, the ion activity product (IAP) equals the solubility product $K_{sp}$. The saturation state is defined as:

$$\Omega = \frac{\text{IAP}}{K_{sp}} = \frac{[\text{Ca}^{2+}][\text{CO}_3^{2-}]}{K_{sp}}$$

Here we use stoichiometric (apparent) $K_{sp}$ values that absorb the activity coefficients, valid at a given temperature, salinity, and pressure.

Step 2: Expressing [CO₃²⁻] in Terms of Measurable Quantities

Since [Ca²⁺] is nearly conservative in seawater ($[\text{Ca}^{2+}] \approx 0.01028 \times S/35$ mol/kg),$\Omega$ is primarily controlled by [CO₃²⁻]. Using the carbonate system:

$$[\text{CO}_3^{2-}] = \frac{\text{DIC} \cdot K_1 K_2}{[\text{H}^+]^2 + K_1[\text{H}^+] + K_1 K_2}$$

Step 3: Substituting to Get $\Omega$ in Terms of pH and DIC

Combining the expressions and noting that $[\text{H}^+] = 10^{-\text{pH}}$:

$$\Omega = \frac{[\text{Ca}^{2+}] \cdot \text{DIC} \cdot K_1 K_2}{K_{sp}\left(10^{-2\text{pH}} + K_1 \cdot 10^{-\text{pH}} + K_1 K_2\right)}$$

Step 4: Sensitivity of $\Omega$ to CO₂ Uptake

When the ocean absorbs CO₂, DIC increases while TA stays constant (CO₂ uptake does not change alkalinity). The net reaction CO₂ + H₂O + CO₃²⁻ → 2HCO₃⁻ consumes CO₃²⁻, so:

$$\frac{d\Omega}{d\,\text{DIC}}\bigg|_{\text{TA}} = \frac{[\text{Ca}^{2+}]}{K_{sp}} \cdot \frac{d[\text{CO}_3^{2-}]}{d\,\text{DIC}}\bigg|_{\text{TA}} < 0$$

The derivative is negative because adding DIC at constant TA decreases [CO₃²⁻] and thus $\Omega$. This is the fundamental mechanism by which ocean acidification threatens calcifying organisms.

Step 5: Depth Dependence via Pressure Effect on K_sp

$K_{sp}$ increases with pressure (depth), because the molar volume of dissolved ions is less than that of solid CaCO₃. The saturation horizon ($\Omega = 1$) occurs where:

$$[\text{Ca}^{2+}][\text{CO}_3^{2-}] = K_{sp}(T, S, P) \quad \text{where} \quad K_{sp}(P) = K_{sp}(0)\exp\left(\frac{-\Delta\bar{V}\,P + 0.5\,\Delta\bar{\kappa}\,P^2}{RT}\right)$$

As atmospheric CO₂ rises and [CO₃²⁻] decreases throughout the water column, the saturation horizon shoals (moves upward), exposing more of the ocean floor to corrosive conditions.

Step 1: Definition of Buffer Capacity

The buffer capacity (buffer intensity) $\beta$ quantifies the resistance of a solution to pH change when acid or base is added. It is defined as the amount of strong acid needed to change pH by one unit:

$$\beta = -\frac{dC_B}{d\,\text{pH}} = \frac{dC_A}{d\,\text{pH}} = 2.303\left([\text{H}^+] + \frac{K_w}{[\text{H}^+]} + \sum_i \frac{C_i K_i [\text{H}^+]}{(K_i + [\text{H}^+])^2}\right)$$

Step 2: Buffer Capacity of the Carbonate System

For the two-proton carbonate system, the buffer contribution involves both dissociation steps. The carbonate buffer intensity is:

$$\beta_{\text{carb}} = 2.303\,\text{DIC}\left(\frac{K_1[\text{H}^+]([\text{H}^+]^2 + 4K_2[\text{H}^+] + K_1 K_2)}{([\text{H}^+]^2 + K_1[\text{H}^+] + K_1 K_2)^2}\right)$$

Step 3: Adding the Borate Buffer

Boric acid (B(OH)₃ / B(OH)₄⁻) provides additional buffering near its pKₐ (~8.6), which is close to ocean pH. The total seawater buffer capacity is:

$$\beta_{\text{total}} = \beta_{\text{carb}} + 2.303 \cdot \frac{B_T K_B [\text{H}^+]}{(K_B + [\text{H}^+])^2} + 2.303\left([\text{H}^+] + \frac{K_w}{[\text{H}^+]}\right)$$

Step 4: Why Buffering Decreases Under Acidification

As CO₂ is absorbed, DIC increases at constant TA, shifting the carbonate speciation toward HCO₃⁻ and away from CO₃²⁻. The buffer factor $\beta_{\text{carb}}$ is maximized when pH is near pK₁ or pK₂, but at ocean pH (~8.1) it lies between these peaks. The change in buffering under CO₂ perturbation is:

$$\frac{d\beta}{d\,\text{DIC}}\bigg|_{\text{TA}} \approx -2.303 \cdot \frac{2[\text{CO}_3^{2-}]}{\text{DIC}} \cdot \beta_{\text{carb}} < 0$$

The buffering capacity decreases as [CO₃²⁻] is consumed, meaning each additional increment of CO₂ causes a progressively larger pH drop. This is the chemical basis for accelerating acidification.

Step 5: Relationship Between Buffer Factor and Revelle Factor

The buffer capacity $\beta$ and the Revelle factor $R$ are related but distinct. The Revelle factor describes pCO₂ sensitivity to DIC, while $\beta$ describes pH sensitivity to acid addition. They are connected through:

$$R = \frac{\text{DIC}}{[\text{CO}_2^*]} \cdot \frac{(\alpha_1 + 2\alpha_2)[\text{H}^+] + S_{\text{TA}}}{S_{\text{TA}}} \qquad \text{where } S_{\text{TA}} = -\frac{\partial \text{TA}}{\partial [\text{H}^+]}$$

High Revelle factor (poor buffering of pCO₂) corresponds to low buffer capacity (poor buffering of pH). Both metrics indicate that the ocean's chemical defense against CO₂ invasion is weakening.

Step 1: Revelle Factor Recalled

The Revelle factor measures how sensitively ocean pCO₂ responds to changes in DIC at constant alkalinity:

$$R = \frac{\text{DIC}}{p\text{CO}_2}\frac{\partial\,p\text{CO}_2}{\partial\,\text{DIC}}\bigg|_{\text{TA}}$$

Step 2: Expressing R in Terms of the DIC/TA Ratio

Using the simplified carbonate alkalinity $A_C \approx \text{TA}$ and defining the ratio $\rho = \text{DIC}/\text{TA}$, the Revelle factor can be shown to depend primarily on this ratio:

$$R \approx \frac{\rho(1 - \rho)^{-1}}{2 - \rho} \cdot \frac{\text{DIC}}{[\text{CO}_2^*]}$$

As $\rho \to 1$ (DIC approaches TA), $R$ increases sharply because nearly all the alkalinity is consumed as bicarbonate, leaving little CO₃²⁻ buffer capacity.

Step 3: Sensitivity of R to Increasing DIC (Climate Feedback)

Differentiating $R$ with respect to DIC at constant TA gives the rate at which buffering deteriorates:

$$\frac{\partial R}{\partial\,\text{DIC}}\bigg|_{\text{TA}} > 0 \qquad \text{always (for ocean-relevant chemistry)}$$

This positive derivative is the mathematical expression of the positive feedback loop: absorbing CO₂ raises DIC, which raises $R$, which makes further absorption less efficient, leaving more CO₂ in the atmosphere.

Step 4: Regional Variation of R

Temperature and alkalinity control the spatial pattern of $R$. Cold waters have higher $K_1$ and $K_2$, shifting speciation toward CO₂* and HCO₃⁻. At high latitudes where $T \approx 2$ degrees C and $\text{TA} \approx 2300$ mumol/kg:

$$R_{\text{polar}} \approx 13\text{-}15 \qquad \text{vs.} \qquad R_{\text{tropical}} \approx 8\text{-}10$$

Polar oceans thus have lower buffering capacity despite being the strongest CO₂ sinks, making them disproportionately vulnerable to acidification.

Step 5: Quantifying the Century-Scale Feedback

From pre-industrial ($p\text{CO}_2 = 280$ ppm) to doubled CO₂ ($560$ ppm), the Revelle factor increases from ~10 to ~14 globally. The fractional uptake efficiency$1/R$ decreases accordingly:

$$\frac{\Delta\text{DIC}/\text{DIC}}{\Delta p\text{CO}_2/p\text{CO}_2} = \frac{1}{R} \quad \Longrightarrow \quad \text{efficiency drops from } \sim 10\% \text{ to } \sim 7\%$$

This means the ocean will absorb a smaller fraction of future CO₂ emissions, amplifying climate warming by 4-14% (the ocean carbon-climate feedback quantified in IPCC AR6).