The exponential function and some of its uses in oceanography

1. The exponential function and some of its uses in oceanography • What is the exponential function? • Three examples of its use in Oceanography • Growth of phytoplankton. • Attenuation of light. • Phytoplankton growth and light.

2. The exponential function: f(x)=Ae(Bx), e=2.71828183… Note: any power function can be translated to exponentials: ax=(eln(a))x= exln(a) • Derivative proportional to the function which means in solve the differential equation:

3. Probabilities and the exponential function: • You have 100 pennies. • Toss them and take out all that are ‘head’. • Keep tossing and write down the number of heads you got per toss. • When done, use Excel plot the number of heads you got as a function of tosses. • Fit an exponential function to the data. What is the exponent of the best fit? Is it sensible? • ln{1/2}=-0.69. • 6. What if you had 100 dice. How would the curve change?

4. Coin toss results:

5. Phytoplankton growth: You have a series of data from a phytoplankton growth experiment (thank you Kate) where chlorophyll fluoresence (chlF) was used as an indicator of phytoplankton biomass. Plot the data of chlF as function of time and ‘best’ fit: P(t)=P(t=0)emt How would you go about calculating the doubling (halving?) time of the phytoplankton? Doubling time: P(t2)=P(t=0)exp(mt2)=2P(t=0) exp(mt2)=2 t2=ln(2)/m. ln(2)~0.7

6. Phytoplankton growth results: Doubling time = 1.08 days

7. Phytoplankton growth: How fast to phytoplankton concentration increase during the spring bloom? 1. GoMOOS data from E01 for 2007. 2. Based on the chlorophyll data estimate the growth rate? µ = 0.2962

8. How does it compare to lab cultures? What may be the reason for agreement/disagreement? The growth rate is much slower than lab cultures. Some reasons for the discrepancy include: limited light/nutrients and/or predation by grazers.

9. Transmission of light: Beer’s law and the exponential function: Measure the intensity of light at the receiver when only water is in the tank. Add drops of dye (such that total drops in the tank= 1, 2, 4, 8, 16, 32). Measure light intensity in each case.

10. Plot the ratio of light intensity with dye divided by light intensity in tap water as function of drops of dye. Could you use your results to predict how many drops of dye are in another tank? Yes! This is the basis of spectroscopy!

11. Scientists use underwater spectrophotometer to investigate the ocean:

12. Light attenuation in the environment: E(l,z)=E(l,0)e(-k(l)z) Blue ocean Coastal ocean Inland pond

13. Basic instrument to measure water quality. The secchi disk: Secci depth: Depth at which disk disappears.

14. Saturated growth: How much carbon is fixed at a given light level? P=Pmax(1-exp(-I/Ik)) Pmax=Pmax(T,N)

15. Summary: • The exponential function can be used to describe many processes in the ocean. Knowing it and how to manipulate it allows one to better predict and understand the environment. • It turns out that ex with x imaginary is simply related to the sine and cosine functions simplifying much the proof of trigonometric identities  facilitate the formulation and study of waves. • If learning by humans followed the model: • Dk/dt=mk • imagine what a little change in knowledge acquisition rate (m) can do to society! • But remember, things can’t keep growing exponentially forever; after a finite time we reach the carrying capacity of the system/environment…

16. Teaching the exponential function Resource: http://faculty.gvsu.edu/goldenj/exponential.html Some highlights The exponential function and probability: http://faculty.gvsu.edu.goldenj.badpenny.html Exploring growth patterns: http://score.kings.k12.ca.us/lessons/growth.html In marine sciences we see the exponential function showing up in many applications: Light decrease with depth in the ocean. Phytoplankton growth when supplied with a given light level of light and nutrients. 3. Nuclear decay.