2.5 Light in the Ocean

Step 1: Consider a thin slab of water

Imagine a thin horizontal layer of ocean water at depth z with infinitesimal thickness dz. As downwelling irradiance $I(\lambda, z)$ passes through this slab, a fraction of the light is removed by absorption and scattering. The amount removed is proportional to both the incident intensity and the thickness of the slab:

$$dI = -K_d(\lambda) \cdot I(\lambda, z) \cdot dz$$

Step 2: Form the differential equation

The negative sign indicates that intensity decreases with increasing depth. Rearranging gives a first-order linear ODE:

$$\frac{dI}{dz} = -K_d(\lambda) \cdot I$$

This states that the rate of light loss per unit depth is proportional to the current irradiance -- a hallmark of exponential decay. The proportionality constant $K_d(\lambda)$ is the diffuse attenuation coefficient.

Step 3: Separate variables and integrate

Separating the variables I and z and integrating from the surface (z = 0) to depth z, with the boundary condition $I(\lambda, 0) = I_0(\lambda)$:

$$\int_{I_0}^{I(z)} \frac{dI'}{I'} = -\int_0^z K_d(\lambda) \, dz'$$

Step 4: Evaluate the integrals

The left side gives a natural logarithm. If $K_d$ is approximately constant with depth (a reasonable assumption for a well-mixed layer), the right side evaluates directly:

$$\ln\!\left(\frac{I(z)}{I_0}\right) = -K_d(\lambda) \cdot z$$

Step 5: Exponentiate to obtain Beer-Lambert law

Taking the exponential of both sides yields the Beer-Lambert law in its standard form:

$$I(\lambda, z) = I_0(\lambda) \exp\!\left(-K_d(\lambda) \cdot z\right)$$

Step 6: Euphotic zone depth

The euphotic zone depth $z_{eu}$ is defined where $I = 0.01 \, I_0$ (the 1% light level). Setting $I/I_0 = 0.01$ and solving:

$$0.01 = e^{-K_d z_{eu}} \quad \Longrightarrow \quad z_{eu} = \frac{\ln(100)}{K_d} = \frac{4.605}{K_d}$$

For clear ocean water ($K_d \approx 0.04$ m$^{-1}$), $z_{eu} \approx 115$ m. For turbid coastal water ($K_d \approx 1.0$ m$^{-1}$), $z_{eu} \approx 4.6$ m.

Step 1: Inherent optical properties (IOPs)

Light attenuation in seawater is governed by two fundamental processes: absorption (coefficient $a$, in m$^{-1}$) and scattering (coefficient $b$, in m$^{-1}$). The beam attenuation coefficient is their sum:

$$c(\lambda) = a(\lambda) + b(\lambda)$$

Step 2: From beam attenuation to diffuse attenuation

The beam attenuation coefficient c applies to a narrow collimated beam (a single ray). In the ocean, sunlight arrives from many directions (a diffuse light field). The diffuse attenuation coefficient $K_d$ describes the decay of the total downwelling irradiance $E_d$ and is defined operationally:

$$K_d(\lambda) = -\frac{1}{E_d} \frac{dE_d}{dz}$$

Step 3: Radiative transfer connection

From radiative transfer theory, the downwelling irradiance involves integrating radiance over the downward hemisphere. The average cosine $\bar{\mu}_d$ characterizes the angular distribution of the downwelling light field:

$$\bar{\mu}_d = \frac{E_d}{E_0^d} = \frac{\int_{2\pi^-} L(\theta,\phi)\cos\theta\, d\Omega}{\int_{2\pi^-} L(\theta,\phi)\, d\Omega}$$

where $E_0^d$ is the downwelling scalar irradiance. For a perfectly collimated beam from zenith, $\bar{\mu}_d = 1$. For a completely diffuse (isotropic) field, $\bar{\mu}_d = 0.5$. In the ocean, $\bar{\mu}_d \approx 0.7\text{--}0.9$.

Step 4: Gershun's equation

Gershun's equation relates the divergence of the net irradiance vector to absorption. For a plane-parallel ocean (properties vary only with depth), this leads to:

$$K_d(\lambda) \approx \frac{a(\lambda) + b_b(\lambda)}{\bar{\mu}_d}$$

Here $b_b$ is the backscattering coefficient (the fraction of scattered light redirected upward). Forward-scattered light continues contributing to the downwelling irradiance, so only backscattering reduces $E_d$. Typically $b_b \ll a$ in the open ocean, so $K_d \approx a/\bar{\mu}_d$.

Step 5: Practical approximation

For the open ocean where scattering is relatively low and the light field is quasi-collimated ($\bar{\mu}_d \approx 0.8$), the diffuse attenuation coefficient is approximately:

$$K_d(\lambda) \approx 1.15 \cdot a(\lambda) + 1.25 \cdot b_b(\lambda)$$

This makes $K_d$ an apparent optical property (AOP) -- it depends on both the water's IOPs and the angular structure of the light field. However, it is relatively stable under typical oceanic conditions and can be measured easily with a profiling radiometer.

Step 1: Visibility contrast threshold

A Secchi disk (white disk, 30 cm diameter) becomes invisible when the contrast between the disk and the surrounding water falls below the human eye's threshold, typically $C_{\min} \approx 0.02$ (Duntley, 1952). The contrast is defined as:

$$C = \frac{L_{\text{disk}} - L_{\text{water}}}{L_{\text{water}}}$$

Step 2: Light path to the disk and back

Light from the sun must travel downward to the disk at depth $z_{SD}$, reflect off the disk, and return upward to the observer. The total optical path length involves attenuation by $K_d$ on the way down and a similar coefficient on the way up. If we approximate both paths using $K_d$:

$$I_{\text{reflected}} \propto I_0 \, R \, \exp(-K_d \, z_{SD}) \cdot \exp(-K_d \, z_{SD}) = I_0 \, R \, \exp(-2 K_d \, z_{SD})$$

where R is the reflectance of the disk (R = 1 for a perfect white disk).

Step 3: Background light (path radiance)

As the observer looks down, scattered light from the water column is added along the viewing path. This path radiance $L_{\text{path}}$ builds up and eventually overwhelms the disk's contrast. The background radiance reaches an asymptotic value determined by backscattering:

$$L_{\text{water}} \propto \frac{b_b}{a + b_b}\left(1 - e^{-2(a + b_b) z_{SD}}\right) \approx \frac{b_b}{K_d \bar{\mu}_d}$$

Step 4: Contrast at the disappearance depth

Setting the contrast equal to the threshold and using simplified radiative transfer, the disk disappears when the two-way attenuation reduces the signal to the noise level. The theoretical analysis by Tyler (1968) and Preisendorfer (1986) yields:

$$C_{\min} \approx R \cdot \exp\!\left(-(K_d + c) \, z_{SD}\right) \cdot \frac{K_d + c}{b_b}$$

Step 5: Empirical simplification

The theoretical expression is complex, but extensive empirical data (Poole and Atkins, 1929; Holmes, 1970) show a remarkably simple and robust relationship. Taking logarithms and using the empirically observed proportionality $c \approx (K_d / 0.4)$ in typical waters:

$$K_d \cdot z_{SD} \approx 1.7 \quad \Longrightarrow \quad K_d \approx \frac{1.7}{z_{SD}}$$

The constant 1.7 (sometimes reported as 1.4 -- 2.0) depends on sky conditions, solar angle, and water type, but 1.7 is the most widely used value. This simple relationship makes the Secchi disk one of the most cost-effective oceanographic instruments.

Step 1: Depth-dependent photosynthesis

Phytoplankton photosynthetic rate (gross primary production) at depth z is proportional to the available light. Using Beer-Lambert attenuation of PAR:

$$P(z) = P_{\max} \cdot \frac{I(z)}{I_0} = P_{\max} \cdot e^{-K_d z}$$

where $P_{\max}$ is the light-saturated photosynthetic rate at the surface (simplified linear model for low light).

Step 2: Depth-independent respiration

Phytoplankton respiration rate R is approximately constant with depth (it does not depend on light). At the compensation depth $z_c$, photosynthesis exactly balances respiration:

$$P(z_c) = R \quad \Longrightarrow \quad P_{\max} \, e^{-K_d z_c} = R \quad \Longrightarrow \quad z_c = \frac{1}{K_d}\ln\!\left(\frac{P_{\max}}{R}\right)$$

Step 3: Sverdrup's key insight -- vertical mixing

In a well-mixed layer of depth $z_m$, phytoplankton are circulated through the entire mixed layer by turbulence. A cell spends equal time at every depth within the mixed layer. The average production experienced by a cell over the mixed layer is the depth-integrated production divided by $z_m$:

$$\langle P \rangle = \frac{1}{z_m}\int_0^{z_m} P_{\max} \, e^{-K_d z}\, dz = \frac{P_{\max}}{K_d \, z_m}\left(1 - e^{-K_d z_m}\right)$$

Step 4: Define the critical depth

The critical depth $z_{cr}$ is the mixed-layer depth at which the depth-averaged production equals the respiration rate. A bloom can occur only when $z_m < z_{cr}$. Setting $\langle P \rangle = R$:

$$\frac{P_{\max}}{K_d \, z_{cr}}\left(1 - e^{-K_d z_{cr}}\right) = R$$

Step 5: Solve for $z_{cr}$ (graphical/transcendental)

Rearranging and using the compensation depth relationship $R/P_{\max} = e^{-K_d z_c}$:

$$\frac{1 - e^{-K_d z_{cr}}}{K_d \, z_{cr}} = e^{-K_d z_c}$$

This is a transcendental equation that must be solved numerically or graphically. However, when $K_d z_{cr} \gg 1$ (the usual case), $e^{-K_d z_{cr}} \approx 0$ and the equation simplifies to:

$$z_{cr} \approx \frac{1}{K_d} \cdot \frac{1}{e^{-K_d z_c}} = \frac{e^{K_d z_c}}{K_d} = \frac{1}{K_d}\cdot\frac{P_{\max}}{R}$$

Step 6: Physical interpretation

The critical depth theory explains the spring bloom in temperate oceans. In winter, deep mixing ($z_m > z_{cr}$) carries phytoplankton below the critical depth, and the population declines because average light is insufficient. In spring, surface warming and reduced winds shoal the mixed layer. When $z_m < z_{cr}$, the bloom initiates:

$$\text{Bloom condition: } z_m < z_{cr} = \frac{1}{K_d}\cdot\frac{P_{\max}}{R}$$

Typical values: $z_{cr} \approx 200\text{--}500$ m in spring at mid-latitudes, while the mixed layer shoals from $\sim 300$ m in late winter to $\sim 20\text{--}50$ m in spring, triggering the bloom.