APPM 2360 Lab 1 Fall 2008

Lab Goals and Instructions

Please read this section carefully. Not following the instructions, including the writing guidelines, can have a very negative impact on your grade!

This lab demonstrates the use of differential equations to model naturally occurring phenomena, specifically the El Niño - Southern Oscillation (ENSO). The purpose of this lab is to use qualitative, quantitative, and analytical methods to study the equations that model the aperiodic warming of the eastern Pacific ocean.

This lab is due in lecture on Friday, October 3. YOU MUST ALSO SUBMIT YOUR PROJECT TO AMESS. Late labs are not accepted. You are encouraged to work in groups of no more than 3 people ( you may work with anyone currently in 2360, and you turn in only copy of the project), though you may work alone.

NOTE: to upload to AMESS your file must be in .pdf format.  The computer labs in the engineering building have a version of Microsoft Word that will allow you to save in .pdf format.  If you want the option to save in .pdf format on your personal version of Word 2007 then you will have to download the additional add in from the Microsoft website. 

On your title page, clearly mark the following information for all group members:

• Names
• Student ID numbers
• Recitation section number, TA, and professor

If you do not include this information, as much as 5% will be deducted from your final score.   Format is worth 20% of your grade.  Please refer to the following writing guidelines for the expository sections of this report:

Writing Guidelines for Labs

You are required to know and follow the Writing Guidelines for all three labs.

Take a look at this sample project and examples of "HOW TO" and "HOW NOT TO" present your results.

Getting help: If you are having trouble using Matlab, some resources can be found on the Differential Equations Lab website. In addition, TA's will be holding regular office hours in the Applied Math Undergraduate Lab (ECCR 143) during the week that the lab is due.

The El Niño Phenomenon

El Niño is the name given to the anomalous warming of the eastern Pacific ocean off the coast of Peru. The peak warming typically occurs in December and so, the Peruvians named it El Niño, Spanish for the Christchild because its arrival coincided with Christmas. The dramatic rise in sea surface temperature (SST) disrupts the normal ecosystem dynamics in the area and the Peruvian anchovy fishing industry collapses. La Niña is the term for the anomalous cooling that sometimes follows an El Niño. Typically the cold SST anomalies during a La Niña are not as intense as the warm SST anomalies during an El Niño. The Southern Oscillation is the name for the changes in atmospheric circulation that accompany El Niño and La Niña. And so the name for the coupled ocean-atmosphere phenomenon is the El Niño - Southern Oscillation or ENSO. Predicting El Niño is important because, when it occurs, it can cause significant changes in global weather patterns. The ENSO cycle has worldwide impacts such as affecting monsoon systems, hurricane frequency and flooding, which can lead to infectious disease epidemics.

Let's look at actual SST data in the tropical Pacific to understand the phases of ENSO. More data shows that the development of ENSO can be traced as waves in the subsurface ocean along the equator in the Pacific.

The Niño 3 index is an average of the sea surface temperatures in the region 150 degrees West - 90 degrees West longitude and 5 degrees North to 5 degrees South latitude. When the index is positive the waters are warmer than normal, when the index is negative, the waters are colder. El Niños occur when the water is much warmer than normal for a sustained period of time. Here is a time series of the Niño 3 index.

Define the variable T(t) to be the Niño 3 temperature anomaly where t is time, measured in years.  We'll consider the following three models of how the Niño 3 index might be evolving:

(1)

(2)

(3)

We will study the models using analytical methods (solutions and stability analysis), qualitative methods (plots and vector fields), and quantitative methods (numerical techniques).  We will evaluate the validity of the three models by comparing with real data, the Niño 3 index.

Questions and Issues to Address

Section 1: Get familiar with ENSO using observations

1. Describe in a sentence or two, the time series of the Niño 3 index.

Using the SST and wind data for the equatorial Pacific in three different Decembers and the three vertical slices of the Pacific Ocean , answer the following questions:

2. Characterize the normal state of sea surface temperature (SST) and wind in the Equatorial Pacific.
3. Explain how the El Niño, and La Niña conditions differ from normal conditions.
4. Where is the warmest pool of subsurface water concentrated before an El Niño? as it develops? during?
5. How does the slope of the thermocline (the sharp gradient in the ocean between the very cold deep water and the warmer water at the surface) change during the development of an El Niño?

Section 2: Models 1 and 2

(1)

6. Classify equation (1) using standard terms (order, linear vs. non-linear, autonomous vs. non-autonomous, homogenous vs. nonhomogenous).
7. Write down the solution to equation (1), given the initial condition T(0)=A (some constant).
8. Using this analytic solution, produce a plot of T(t) on t in [0,50] for A=1.0 and for k=0.05,0.03,-.03,and -0.1. These should all be in one figure with a legend indicating the k value for each curve.  How does the solution depend on k?
9. Find any equilibrium solutions of equation (1) and classify their stability. What is the long term behavior of the system?
10. Compare the solution curves of equation (1) to the Niño 3 index data. Do you think that equation (1) is a good model for ENSO? Explain your answer.



(2)



11. Classify equation (2) in standard terms. From now on you may assume that k > 0 and b > 0.
12. Write down the equilibrium solutions for equation (2) and classify their stability. What is the long term behavior of the system?
13. Rescale both time and temperature using the substitutions: = kt and U()= T(t)*(k/b)^-1/2, where U is a function of and T is a function of t. Substitute these terms into equation (2) and eliminate the parameter (b and k) dependence in the equation. Be careful with the left hand side as time is in the derivative term, dT/dt.  Report the steps you take in an appendix.  For information on appendix format, see the writing guidelines.
14. Find the analytical solution of the transformed equation. You can leave it in implicit form.  Show the steps in the appendix and put the solution in your text.
15. What are the equilibria of the transformed equation?  (They should not depend on the parameters k and b.) Classify the stability of the equilibria. What does this analysis tell you about the long term behavior of the system?
16. Using appropriate computer software, plot the vector field of the transformed equation.  Add to the vector field solutions for various initial conditions. Be sure to have exactly one initial condition in each of the following ranges:
    • U(0) < -1
    • -1 < U(0) < 0
    • 0 < U(0) < 1
    • U(0) > 1

    Hint: See Worksheet 2 on the 2460 website to see how to do this using Matlab.

17. Do the plots support your stability analysis? Why or why not?
18. Compare the solution curves of the transformed equation to the Niño 3 index data. Do you think that this equation is a good model for ENSO? Explain your answer.

Section 3: Model 3

Now we modify equation (2) by adding the term - T (t - )

(3)

Equation (3) is the "delayed action oscillator" because the new term involves the temperature at a previous time, t - . This type of ODE which involves a delayed response is known as a delay differential equation. There are many other systems besides ENSO which can be described by delay differential equations.

Equation (3) is complicated enough that we will no longer attempt to find an analytic solution.  The MatLab code dao.m will do the numerical work in this section. Do not change any of the code in the file. For example, say you want a solution from t = 0 to t = 10 for = 15 and = 0.5. Then you would type in the command window:     >> [t,T]=dao(15,0.5,10)

More information about the guts of dao.m can be viewed with the command:     >> help dao

19. (For parts 19 - 22, use = 0.5.) Describe the changes that you observe in the time series as you change . It may help to zoom in on the time series in the MatLab figures. You can do this by clicking on the magnifying glass icon at the top of the MatLab figures and then using the cursor to draw a box around the area on the graph that you want to magnify.
20. You should find through your experiments, a critical value _c that causes a change in the LONG TERM behavior of the system. That is, for < _c the behavior of the system is markedly different from the behavior when > _c. Find an approximate value for _c or a small range of values that contains _c. 
21. Include plots of solutions for alpha values above and below _c.  Describe the end behavior for solutions with  > _c and with  < _c. Which is a better model for the Niño 3 index time series? Why?
22. Now use = 4.0, and run the model for a few different values of . How does the longer delay affect the solutions? How do the solutions compare to the Niño 3 index data?
23. Which solutions (of all the cases you have run) look most like the real data? In what ways do they differ from the real data? Suggest how this model might be improved.

Last modified: Sept. 2008 by Jason Hammond