Problem 91 - Calculus for Biology and Medicine

Question

The absorption of light in a uniform water column follows an exponential law; that is, the intensity \(I(z)\) at depth \(z\) is $$ I(z)=I(0) e^{-\alpha z} $$ where \(I(0)\) is the intensity at the surface (i.e., when \(z=0)\) and \(\alpha\) is the vertical attenuation coefficient. (We assume here that \(\alpha\) is constant. In reality, \(\alpha\) depends on the wavelength of the light penetrating the surface.) (a) Suppose that \(10 \%\) of the light is absorbed in the uppermost meter. Find \(\alpha\). What are the units of \(\alpha\) ? (b) What percentage of the remaining intensity at \(1 \mathrm{~m}\) is absorbed in the second meter? What percentage of the remaining intensity at \(2 \mathrm{~m}\) is absorbed in the third meter? (c) What percentage of the initial intensity remains at \(1 \mathrm{~m}\), at 2 \(\mathrm{m}\), and at \(3 \mathrm{~m} ?\) (d) Plot the light intensity as a percentage of the surface intensity on both a linear plot and a log-linear plot. (e) Relate the slope of the curve on the log-linear plot to the attenuation coefficient \(\alpha\). (f) The level at which \(1 \%\) of the surface intensity remains is of biological significance. Approximately, it is the level where algal growth ceases. The zone above this level is called the euphotic zone. Express the depth of the euphotic zone as a function of \(\alpha\). (g) Compare a very clear lake with a milky glacier stream. Is the attenuation coefficient \(\alpha\) for the clear lake greater or smaller than the attenuation coefficient \(\alpha\) for the milky stream?

Step-by-Step Solution

Answer

(a) \(\alpha = -\ln(0.9)\) with units \(\text{m}^{-1}\); (b)-(e) see solutions; (f) Euphotic depth \(z=\frac{\ln(0.01)}{-\alpha}\); (g) \(\alpha\) is smaller for the clear lake.

Step 1: Understanding the Problem

We are given the function \(I(z) = I(0) e^{-\alpha z}\), where \(I(z)\) is the intensity at depth \(z\), and \(\alpha\) is the attenuation coefficient. We need to find \(\alpha\) given that 10% of light is absorbed in the first meter.

Step 1: Determine Light Intensity at 1 Meter

Since 10% of the light is absorbed in the uppermost meter, 90% remains at \(z = 1\). Thus, \(I(1) = 0.9 \times I(0)\).

Step 2: Set Up the Equation for \(\alpha\)

Using the equation \(I(1) = I(0) e^{-\alpha}\), substitute \(I(1) = 0.9 I(0)\) to get \(0.9 I(0) = I(0) e^{-\alpha}\).

Step 3: Solve for \(\alpha\)

Cancel \(I(0)\) from both sides to get \(0.9 = e^{-\alpha}\). Taking the natural logarithm on both sides, \(-\alpha = \ln(0.9)\). Thus, \(\alpha = -\ln(0.9)\).

Step 4: Determine the Units of \(\alpha\)

The unit of \(\alpha\) should be such that the exponent in the exponentials is unitless. Therefore, since \(z\) is in meters, \(\alpha\) has units of \(\text{m}^{-1}\).

Step 5: Find Absorption in Subsequent Meters

For \(z = 2\), initially, the intensity is \(I(2) = I(1) e^{-\alpha}\). The percentage absorbed in the second meter is \(1 - e^{-\alpha}\). Repeat this for \(z = 3\).

Step 6: Calculate Intensity Remained at Depths

Using \(I(z) = I(0) e^{-\alpha z}\), calculate the percentage remaining at 1m, 2m, and 3m. Substitute \(\alpha\) and find the percentages.

Step 7: Plot the Intensity

Create two plots. A linear plot with depth on the x-axis and intensity percentage on the y-axis, and a log-linear plot to show the exponential decay in a linear fashion.

Step 8: Determine the Slope on Log-Linear Plot

The slope on a log-linear plot \(\ln(I(z))\) vs. \(z\) is \(-\alpha\). This shows the rate of light attenuation per unit depth.

Step 9: Calculate Euphotic Zone

The euphotic zone ends when only 1% of \(I(0)\) remains, \(0.01 = e^{-\alpha z}\). Solve for \(z\) to find \(z = \frac{\ln(0.01)}{-\alpha}\).

Step 10: Compare Attenuation in Water Bodies

In a clear lake, \(\alpha\) is small implying less light absorption, whereas in a milky stream \(\alpha\) is larger due to more scattering and absorption. Thus, \(\alpha\) is greater for a milky stream.

Key Concepts

Attenuation Coefficient

The attenuation coefficient, denoted by \(\alpha\), is a crucial concept in understanding how light diminishes as it travels through a medium like water. In the equation \(I(z) = I(0) e^{-\alpha z}\), \(\alpha\) serves as the rate at which light intensity decreases with depth \(z\).
In practical terms, \(\alpha\) measures the effectiveness of absorption and scattering of light in water. It tells us how quickly light fades away as we move away from the source.
The units of the attenuation coefficient are \text{m}^{-1} because it indicates how much light diminishes per meter.
Understanding \(\alpha\) helps differentiate between various types of water bodies:

• For clear water bodies like lakes, \(\alpha\) is small, indicating minimal loss of light and high water transparency.
• In murkier waters, such as glacier streams with sediment, \(\alpha\) is larger, reflecting higher rates of light absorption and scattering.

These differences explain why visibility and habitats for aquatic organisms significantly vary based on water clarity.

Euphotic Zone

The euphotic zone is a biological concept stemming from the interactions between light penetration and aquatic life. It refers to the uppermost part of a body of water where sunlight is sufficient for photosynthesis, enabling plant and algal growth.
As light penetrates water, it diminishes according to the exponential decay function. The euphotic zone ends at the depth where only 1% of surface light remains. Mathematically, it is defined by the equation:

• For a given attenuation coefficient \(\alpha\), the depth \(z\) of the euphotic zone can be found using:\[z = \frac{\ln(0.01)}{-\alpha}\]

This zone is crucial for ecology because it is the region where most marine life productivity occurs, driven by sunlight.
In clearer waters, the euphotic zone extends deeper, supporting more extensive aquatic life. Conversely, in water with high attenuation like murky streams, the euphotic zone is shallow due to high rates of light absorption and scattering.

Light Intensity Function

The light intensity function defines how light intensity decreases with depth in water. It is expressed as \(I(z) = I(0) e^{-\alpha z}\), where \(I(z)\) is the intensity at depth \(z\) and \(I(0)\) represents the surface intensity.
This function is linked to exponential decay, meaning light diminishes rapidly as we move deeper. The rate of decrease is dictated by the attenuation coefficient \(\alpha\).
There are several practical applications of this formula:

• Determining the percentage of light intensity at any depth. For instance, at 1 meter, if 10% absorption occurs, 90% remains.
• Comparing how light behaves in different water conditions to infer water quality.

The graphical representation of the intensity function often utilizes both linear and log-linear plots to illustrate how quickly light declines. In a log-linear graph, the slope directly relates to \(-\alpha\), offering a visual of the attenuation rate.
This function is integral in environmental and oceanographic studies as it helps model light availability and subsequent ecological implications in aquatic systems.