4.1 Marine Ecosystems

Step 1: Define Production and Respiration Components

Gross Primary Production (GPP) is the total carbon fixed by photosynthesis. Net Primary Production (NPP) subtracts autotrophic respiration R_a. Net Ecosystem Production (NEP) further subtracts heterotrophic respiration R_h:

$$\text{GPP} = \text{NPP} + R_a, \qquad \text{NEP} = \text{NPP} - R_h = \text{GPP} - R_a - R_h$$

Step 2: Light-Dependent Production with Depth

Light attenuates exponentially with depth following Beer-Lambert law. The photosynthesis rate at depth z using the Webb (1974) formulation is:

$$P(z) = P_{\max}\!\left(1 - e^{-\alpha I_0 e^{-K_d z}/P_{\max}}\right)$$

where I(z) = I₀ exp(-K_d z) is the irradiance at depth z, K_d is the diffuse attenuation coefficient, alpha is the photosynthetic efficiency, and P_max is the maximum photosynthetic rate.

Step 3: Compensation Depth from P(z) = R

At the compensation depth z_c, photosynthesis equals respiration for a single cell. Setting P(z_c) = R and solving for z_c:

$$P_{\max}\!\left(1 - e^{-\alpha I_0 e^{-K_d z_c}/P_{\max}}\right) = R$$

$$z_c = -\frac{1}{K_d}\ln\!\left[-\frac{P_{\max}}{\alpha I_0}\ln\!\left(1 - \frac{R}{P_{\max}}\right)\right]$$

Step 4: Sverdrup Critical Depth Integral

Sverdrup (1953) defined the critical depth z_cr as the depth where depth-integrated production equals depth-integrated community respiration R_c over the mixed layer. A bloom can develop only when the mixed layer depth (MLD) is shallower than z_cr:

$$\int_0^{z_{cr}} P(z)\,dz = \int_0^{z_{cr}} R_c\,dz = R_c \cdot z_{cr}$$

Step 5: Evaluate the Integral for Critical Depth

For the simplified linear P-I model P(z) = alpha I₀ exp(-K_d z) (valid when light is not saturating), the integral evaluates analytically:

$$\int_0^{z_{cr}} \alpha I_0 e^{-K_d z}\,dz = \frac{\alpha I_0}{K_d}\left(1 - e^{-K_d z_{cr}}\right) = R_c \cdot z_{cr}$$

This transcendental equation is solved numerically for z_cr. The bloom condition is then: MLD < z_cr, which is satisfied when the mixed layer shoals in spring due to warming and reduced wind mixing.

Step 1: Trophic Transfer Efficiency

Energy flows from one trophic level to the next with an efficiency epsilon. At each level, energy is lost to respiration, excretion, and egestion. The transfer efficiency from trophic level n to n+1 is:

$$\epsilon = \frac{P_{n+1}}{P_n} = \frac{\text{Production at level } n{+}1}{\text{Production at level } n} \approx 0.10$$

Step 2: Production at Trophic Level n

Starting from NPP at the base, the production at trophic level n is a geometric decay. This is the "10 percent rule" of Lindeman (1942):

$$P_n = \text{NPP} \cdot \epsilon^{n-1}$$

Step 3: Energy Budget for a Single Trophic Level

The energy budget for any organism partitions ingested energy (I) among assimilation (A), egestion (F), production (P), and respiration (R):

$$I = F + A, \qquad A = P + R$$

$$\Rightarrow \quad I = P + R + F, \qquad \text{Gross growth efficiency} = \frac{P}{I} = \frac{A - R}{I}$$

Step 4: Export Production and the Biological Pump

The fraction of NPP that sinks below the euphotic zone as export production (EP) is related to the f-ratio. The efficiency of the biological pump in sequestering carbon to depth z is:

$$\text{EP} = f \cdot \text{NPP}, \qquad \text{Transfer efficiency}(z) = \frac{F(z)}{F(z_0)} = \left(\frac{z}{z_0}\right)^{-b}$$

Only ~1% of surface NPP reaches the deep sea floor (4000 m), with most organic matter remineralized in the upper 1000 m (mesopelagic zone).