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APPM 2360 Lab 1
Fall 2008 El Nino - Southern Oscillation (ENSO)
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 Nino - 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. Late labs are not accepted. You are encouraged to work in groups of no more than 3 people, though you may work alone.
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:
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.
EI Nino 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 EI Nino, Spanish for the Christchild
"use its arrival coincided with Christmas. The dramatic rise in sea surface temperature (SST) disrupts the Jrmal ecosystem dynamics in the area and the Peruvian anchovy fishing industry collapses. La Nina is the term for the anomalous cooling that sometimes follows an El Nino. Typically the cold SST anomalies during a La Nina are not as intense as the warm SST anomalies during an El Nino. The Southern Oscillation is the name for the changes in atmospheric circulation that accompany El Nino and La Nina. And so the name for the coupled ocean-atmosphere phenomenon is the El Nino - Southern Oscillation or ENSO. Predicting El Nino 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 EN SO can be traced as waves in the subsurface ocean along the equator in the Pacific. The Nino 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 Ninos occur when the water is much warmer than normal for a sustained period of time. Here is a time series of the Nino 3 index. Defme the variable T(t) to be the Nino 3 temperature anomaly where t is time, measured in years. We'll consider the following three models of how the Nino 3 index might be evolving:
dT =kT
dt
-=kT-bT
dT
.
3
dt
dtT(t)
d
= T(t)-
~(t)
- aT(t - if)
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 Nino 3 index.
Questions and Issues to Address
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. 3. 4. 5. Characterize the normal state of sea surface temperature (SST) and wind in the Equatorial Pacific. Explain how the El Nino, and La Nina conditions differ from normal conditions. Where is the warmest pool of subsurface water concentrated before an El Nino? as it develops? during? 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 Nino?
L.Cv~t~
L 0~
'd \ -\-v-.-e.
dT = kT
dt
6. ClassifY equation (I) using standard terms (order, linear vs. non-linear, autonomous non-autonomous, vs. homogenous vs. nonhomogenous). 7. Write down the solution to equation (I), given the initial condition T(O)=A (some constant). <)Using this analytic solution, produce a plot ofT(t) on t in [O,50j 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? \.2..IFind any equilibrium solutions of equation (I) an classifY their stability. What is they; terliihavio09 ~stem2 . 1oOC<rm e t~ solution curves of equation (1) to the Nino 3 index data. Do you think that equation (I) is a good model for ENSO? EXRlain your answer.
dt
dT = kT- bT3
11. ClassifY equation (2) in standard terms. From now on you ma assum t > 0 and b > O. ~ Write down the equilibrium solutions for equation (2) and lassifY their stability. hat is the long term behavior of the system? both time and temperature using the substitutions: T = kt and U( T)= T(t)*(k/ l/2, where U is a function OfT and T is a function oft. 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. Find the analytical solution of the transformed equation. You can leave it in implicit form. rShowthe steps in the appendix and Rut the solution in your text. v L-t);::;0 0D\\Je \.J15. What are the equilibria of the transformed equation? (They should not dtpend on the parameters k and b.) ClassifY the stability ..2fthe eguilibrj~. What does this analysis tell you about the long term behavior of the system? (J]> Using appropriate computer software, plot the vector field ofthe 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: o U(O) <-1 o -1 < U(O) < 0 o O<U(O)< 1 o U(O) > 1
@ Rescale
br
@
,,t -
,or
17. Do the plots support your stability analysis? Why or why not? 18. Compare the solution curves of the transformed equation to the Nino 3 index data. Do you think that this equation is a good model for ENSO? Explain your answer.
dtT(t) = T(t) - ~(t) - aT(t - 5)
Equation (3) is the "delayed action oscillator" because the new term involves the temperature at a previous time,
d
t - 6. 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 fmd 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
[t,
a=
15 and 6 = 0.5. Then you would type in the command window:
>>
T] =dao (15,0,5,10)
19. (For parts 19 - 22, use Ii = 0.5.) Describe the changes that you observe in the time series as you change Q. 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
Q
c that causes a change in the LONG TERM
behavior of the system. That is, fora < Qc the behavior of the system is markedly different from the behavior when
a >ac.
Find an approximate value for ac or a small range of values that contains
Q c.
Q
c.
(~'
21. Include plots of solutions for alpha values above and below with
Describe the end behavior for solutions
\
a > ac
and with 0 < 0c. Which is a better model for the Nifio 3 index time series? Why?
22. Now use 6 = 4.0, and run the model for a few different values of a. How does the longer delay affect the solutions? How do the solutions compare to the Nifio 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. Source for model: "A Delayed Action Oscillator for ENSO" by Suarez and Schopf (Journal of the Atmospheric Sciences, 45, 1988, pp. 3283-3287)