The exponential function and some of its The exponential function and some of its uses in oceanographyuses in oceanography

•What is the exponential function?What is the exponential function?

•Three examples of its use in OceanographyThree examples of its use in Oceanography

•Growth of phytoplankton.Growth of phytoplankton.

•Attenuation of light.Attenuation of light.

•Phytoplankton growth and light.Phytoplankton growth and light.

The exponential function:The exponential function:

f(x)=Aef(x)=Ae(Bx)(Bx), e=, e=2.71828183…2.71828183…

Note: any power function can be Note: any power function can be translated to exponentials:translated to exponentials:

aaxx=(e=(eln(a)ln(a)))xx= = eexln(a)xln(a)

•Derivative proportional to the function Derivative proportional to the function which means in solve the differential which means in solve the differential equation: equation:

....!32

132

++++=xx

xex

( ) ( )xBfdx

xdf=

Probabilities and the exponential function:Probabilities and the exponential function:

You have 100 pennies.You have 100 pennies.

1.1. Toss them and take out all that are ‘head’.Toss them and take out all that are ‘head’.

2.2. Keep tossing and write down the number of Keep tossing and write down the number of heads you got per toss.heads you got per toss.

3.3. When done, use Excel plot the number of When done, use Excel plot the number of heads you got as a function of tosses.heads you got as a function of tosses.

4.4. Fit an exponential function to the data. Fit an exponential function to the data. What is the exponent of the best fit? Is it What is the exponent of the best fit? Is it sensible?sensible?

5.5. ln{1/2}=-0.69.ln{1/2}=-0.69.

6. What if you had 100 dice. How would the 6. What if you had 100 dice. How would the curve change?curve change?

Probabilities and the Exponential Function

y = 92.634e-0.6586x

R2 = 0.9947Group 1

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0 1 2 3 4 5 6 7 8

Toss #

# of Heads

Probabilities and the Exponential Function

y = 98.185e-0.6612x

R2 = 0.9547Group 2

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Toss #

# of heads

Coin toss results:Coin toss results:

Phytoplankton growth:Phytoplankton growth:

How would you go about calculating the How would you go about calculating the doubling (halving?) time of the doubling (halving?) time of the phytoplankton?phytoplankton?

Plot the data of chlPlot the data of chlFF as function of time as function of time

and ‘best’ fit:and ‘best’ fit:P(t)=P(t=0)eP(t)=P(t=0)ett

Doubling time:Doubling time:P(tP(t22)=P(t=0)exp()=P(t=0)exp(tt22)=2P(t=0))=2P(t=0)

exp(exp(tt22)=2 )=2 tt22=ln(2)/=ln(2)/ln(2)~0.7ln(2)~0.7

You have a series of data from a You have a series of data from a phytoplankton growth experiment (thank you phytoplankton growth experiment (thank you Kate) where chlorophyll fluoresence (chlKate) where chlorophyll fluoresence (chlFF) )

was used as an indicator of phytoplankton was used as an indicator of phytoplankton biomass.biomass.dayday 11 22 66 88 1010

ChlChlFF 0.340.34 1.171.17 16.516.533

30.130.1 37.437.4

Phytoplankton Growth

y = 0.3386e0.5305x

R2 = 0.9315

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Days

Chlorophyll Fluoresence

Phytoplankton growth results:Phytoplankton growth results:

Doubling time = 1.08 daysDoubling time = 1.08 days

Phytoplankton growth:Phytoplankton growth:

How fast to phytoplankton concentration How fast to phytoplankton concentration increase during the spring bloom?increase during the spring bloom?

1. GoMOOS data from E01 for 2007.1. GoMOOS data from E01 for 2007.

2. Based on the chlorophyll data estimate the 2. Based on the chlorophyll data estimate the growth rate? µ = 0.2962growth rate? µ = 0.2962

Phytoplankton Growth rate (GOMOOS data)

y = 0.8377e0.2962x

R2 = 0.9564

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14

0 1 2 3 4 5 6 7 8 9 10

Day

Chlorophyll

How does it compare to lab cultures? What How does it compare to lab cultures? What may be the reason for may be the reason for agreement/disagreement?agreement/disagreement?

The growth rate is much slower than lab The growth rate is much slower than lab cultures. Some reasons for the discrepancycultures. Some reasons for the discrepancyinclude: limited light/nutrients and/orinclude: limited light/nutrients and/orpredation by grazers. predation by grazers.

Transmission of light:Transmission of light:

Beer’s law and the exponential function:Beer’s law and the exponential function:

Measure the intensity of light at the receiver when Measure the intensity of light at the receiver when only water is in the tank.only water is in the tank.

Add drops of dye (such that total drops in the tank= 1, Add drops of dye (such that total drops in the tank= 1, 2, 4, 8, 16, 32). Measure light intensity in each case.2, 4, 8, 16, 32). Measure light intensity in each case.

Could you use your results to predict how Could you use your results to predict how many drops of dye are in another tank?many drops of dye are in another tank?Yes!Yes!

This is the basis of spectroscopy!This is the basis of spectroscopy!

Plot the ratio of light intensity with dye divided by Plot the ratio of light intensity with dye divided by light intensity in tap water as function of drops of light intensity in tap water as function of drops of dye.dye.

Transmission of Light

y = 952.05e -0.1678x

R2 = 0.9123

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1400

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# of drops

Light Intensity

Scientists use underwater spectrophotometer to investigate the ocean:

Light attenuation in the environment:E(,z)=E(0)e(-k()z)

Blue ocean

Coastal ocean

Inland pond

Basic instrument to measure water quality. The secchi disk:

Secci depth: Depth at which disk disappears.

Saturated growth:

How much carbon is fixed at a given light level?

P=Pmax(1-exp(-I/Ik))

Pmax=Pmax(T,N)

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=k

imagine what a little change in knowledge acquisition rate () 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…

Teaching the exponential functionTeaching the exponential function

Resource: Resource: http://faculty.gvsu.edu/goldenj/exponential.html

Some highlightsSome highlights

The exponential function and probability:The exponential function and probability:http://faculty.gvsu.edu.goldenj.badpenny.html

Exploring growth patterns:Exploring growth patterns:http://score.kings.k12.ca.us/lessons/growth.html

In marine sciences we see the exponential function showing up In marine sciences we see the exponential function showing up in many applications:in many applications:

1.1. Light decrease with depth in the ocean.Light decrease with depth in the ocean.2.2. Phytoplankton growth when supplied with a given light Phytoplankton growth when supplied with a given light

level of light and nutrients.level of light and nutrients.3. Nuclear decay. 3. Nuclear decay.