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Calculations Using Beer's Law

The decrease in the intensity of light as it travels through the water column is called light attenuation. Light attenuation is caused by the combined absorption and scattering properties of everything in the water column, including the water itself. In most ocean waters (and for the purposes of this class) light decreases as a function of depth in the water column in a way that can be described mathematically. The mathematical description of light attenuation in the water column is known as Beer's Law.

Beer's Law tells us that light decreases exponentially with depth. An exponential relationship is described by a curved line; human population growth can be described by an exponential curve. What this means to us no-math-heads is that light decreases very rapidly near the surface and decreases more slowly as we go deeper.

Beer's Law as a math equation looks like this:

I z = I 0 * exp -k*z

Hey, that's not too complicated, is it? But what the heck does it mean?

I z (pronounced eye-sub-zee) is a symbol that stands for the Intensity of light at depth z. (Remember that oceanographers use the z-axis for depth? If not, go review Use of Graphs in Section II). I 0 (pronounced eye-sub-zero) is a symbol that stands for the Intensity of light at depth zero (0). Depth zero is the surface. Depth z is any depth below the surface.

Notice that z appears in the right-hand side of the equation as a superscript. Z still stands for depth and what this equation tells us in words is that the light intensity at a particular depth, z, is equal to the light intensity at the surface multiplied by the exponent (which is 2.71828) raised to the minus k times z power. (The * symbol means to multiply.) Okay, that last part is tricky but any good calculator will get you through it without knowing what it means. But for those of you who do care, a negative exponent is the same thing as the inverse of that number, so if we divide 1 by exp k* z we can get the same answer.

The k symbol in this equation is known as the attenuation (or extinction) coefficient. It describes how quickly light attenuates or "goes extinct" in a particular body of water. Just using your gut feelings, what would you expect about light trying to penetrate a water column with high values of k, in other words, a water column with lots of attenuation or extinction? If you guessed that light would diminish rapidly, then you are on the right track. If you didn't guess this answer, then think about it a little bit more.

We can use Beer's Law to answer this question about the effects of k on the depth of the euphotic zone (i.e. whether light penetrates deeply or diminishes rapidly). First, consider the case where depth, z, is zero (0), i.e. at the surface. What is the value of I z?

First, compute k * z. If z equals zero (0), then k * z is zero (0). (See, it's not so hard.)

If k * z is zero, then exp -k*z equals one (1) because any number raised to the zero (0) power is one, by the authority of the High Council on Mathematics (and because all the math books tell us that).

So, if exp -k*z equals one (1) then I z = I 0 and we are done. The intensity of light at the surface equals the intensity of light at the surface.

I'll admit that is not a shocking conclusion and it sure seems like a lot of mental loop-dee-loops to figure out something obvious, but what I want you to get out of this is a willingness to understand math. If you are willing to give this simple math a try, then you will gain confidence in your math abilities and lose the fear of math that pesters us worse than that guy in our dreams with the funny fingernails.

Give Beer's Law a try with real numbers and see how it operates under different values for k.  Here's a table for you to fill out using Beer's Law. Use a surface light intensity (I 0) of 1500 microEinsteins per square meter per second and compute I z for each depth listed.

Depth
(meters)
k=0.01 k=0.04 k=0.8 k=0.12
0
20
40
60
80
100
120
140
160
180
200

Mark the approximate depth of the euphotic zone in each case, i.e. where I z equals 1% of I 0?

How do values of k change the depth of the euphotic zone?