The El Niño / Southern Oscillation
For many years, fishermen and coastal residents of Peru noticed that conditions in the ocean west of their country varied widely from year to year. In most years, the waters were cool, but were warmed slightly during the early part of a year. However, some years the warming trend would begin much earlier, and would be much stronger. This led to anomalous natural events, such as heavy rains in desert lands, and changes in migration patterns of sea life. At times sea life was even killed by dramatic temperature changes. The phenomenon came to be called El Nino "the boy"  by the fishermen , for the way it coincided with the advent season. El Niño has been associated with significant changes in weather as far away as the North America, where it is linked to increased rainfall and warmer temperatures in coastal states.
Study by scientists has elaborated on the understanding of the phenomenon. El Niño is known by scientists as the "El Niño  Southern Oscillation" (ENSO). The essential features of the phenomenon include a drop in the trade winds across the south central Pacific Ocean, coupled with a related increase in the ocean current pushing the warmer water of the western Pacific toward the eastern part of the ocean. The Southern Oscillation of the appellation is due to the work of a scientist named Gilbert Walker. He noticed that when pressure was high in northern Australia, then it was low near Tahiti. Conversely, high pressure near Tahiti is coupled with low pressure in northern Australia. In short, the phenomenon is a dramatic manifestation of the functioning of the Pacific Ocean, and the air over it, as a single connected meteorological system.
Under ordinary circumstances, these flow from east to west during summer in the southern hemisphere. This has the effect of raising the sea level in the western Pacific, and of keeping the warmer water there from flowing toward South America. Instead, it stays more or less in place and is heated. During an ENSO year, the velocity of the winds is severely reduced, and sometimes even reversed. This allows the warmer water of the western Pacific to flow eastward, driving up temperatures off the coast of South America. The effects of this water temperature rise are translated to the air, and thereby to the entire West Coast of the Americas. For more qualitative details of ENSO, consult sites at the National Center for Atmospheric Research, or at the National Oceanic and Atmospheric Administration.
ENSO comes erratically  so erratically that it has thus far defied longterm prediction. The very fact that one refers to ordinary years is evidence that one does not expect to experience ENSO very often. One might attempt to describe this sort of behavior purely in statistical terms, but if one does so, no insight is gained into the causes of the phenomenon. On the other hand, if one can form a mathematical model that explains the qualitative features of ENSO, then one might be able at least to understand some of the science behind it. A paper in Science [1] has attempted to explain the qualitative features of ENSO using a simple differential equations model. This model is the subject of this exercise.
Imagine the equatorial Pacific Ocean to be a box (indeed, a rather large box) of fluid. One feature of this area of the Pacific is that the water is divided roughly into two zones in the vertical direction. There is a surface body of water that experiences changing temperature, and in which the currents flow, as well as a deeper body of water that remains at a relatively constant temperature. The two bodies are separated by a surface known as the thermocline. We denote the constant temperature beneath the thermocline by T_{}. There is a second equilibrium with which we must be concerned. Each day the sun shines on the ocean and heats it. At the same time, the ocean radiates and conducts heat away from itself, into the atmosphere. There is some average temperature, which we denote T*, that the ocean surface maintains (approximately) in this process. If the ocean temperature rises above T*, then it will cool according to Newton's Law of cooling:
T ' = A ( T  T* )
Problem 1: Solve the equation above for T. Show that this model does predict that if T > T* then the temperature T tends to drop, while if T < T* then the temperature tends to rise. If the system were left alone for a very long time, what would happen to the temperature? What is the role of the parameter A?
In the Pacific Ocean, we must also take the currents we have discussed into account, since they transport bodies of water from one part of the ocean to another. This water carries heat with it, providing a convective component to the temperature.
We concern ourselves with only one spatial dimension  denoting the eastwest breadth of the ocean. The features we wish to model are the temperatures above the thermocline in the east, T_{e}, and in the west, T_{w}, as well as the speed u of the current. The current u is taken to be positive when it moves from west to east. The easterly trade winds push a certain average current toward the east, which we denote u* <= 0. This current must effectively be overcome if ENSO is to occur.
Differences in temperature from east to west alter the trade winds, which in turn alters the current. On the other hand, the faster the current, the more friction will tend to damp it. Thus we write an equation for the current of the form
u' = ( B / )( T_{e} T_{w })  C(u  u*)
The constants B and C are positive. B describes the rate of flow of current due to a temperature differential between east and west, while C describes a frictional resistance to flow of current.
Problem 2: Suppose that T_{e }and T_{w }are constant, with T_{e }< T_{w}. What is the equilibrium current that results from this state? Discuss what can happen when T_{e }and T_{w }are very close together (or equal), and when they are far apart. What happens if there is no frictional resistance to the current ( C=0 )?
The temperature depends on several factors. It is affected by Newton's law of cooling, as discussed in Problem 1, however the presence of the current complicates the situation. In the equations below, the first term accounts for the current, while the second you will recognize from Newton's Law. We set T_{}=0, in order to simplify calculations. If actual temperatures are desired, we may simply add the actual deepwater temperature to the results.
T_{w}' =  ( uT_{e} / )  A( T_{w}  T* )
T_{e}' = ( uT_{w} / )  A( T_{e}  T* )
In short, the first equation says that the temperature in the west is increased when eastern water flows into the region (remember that u is negative for an easterly current), and is otherwise affect by natural cooling (or heating) according to Newton's law. The second is similar. The constant .denotes the width of the ocean.
Problem 3: Solve the equations for T_{e} and T_{w} in case A=0 and u=u*, Describe the behavior of the solution. To what does this behavior correspond in the physical situation?
Problem 4: Compute the equilibria of the equations, and their stability. For simplicity, use u* = 0. Pay special attention to the case when
2B > (4A + C) C^{2} / [ T*(C2A) ]
Plausible values for the parameters may be found in [1]. They are given in the table below.
A 
B 
C 
T* 
u* 

1 
663 
3 
12 
14.2 
14 
The unit of time is taken to be years, and the unit of distance is the megameter (one million meters, or 1000 km).
Problem 5: Use a numerical
differential equation solver to compute and plot u and the
difference against time t. Use the values in the table
above, except for u* and B. Use u* = 0, and
take B = 100, 350 and 700. Run the simulation from t=0
to t=30. Use an initial condition of u = 10, T_{w}=14,
and T_{e}=10. Interpret the results
in view of Problem 4.
Problem 6: Do Problem 5 again using u* = 14.2. Interpret the results.
The results you see for the largest parameter in Problems 5 and 6 typify a chaotic solution to the differential equation. Without giving a technical definition for the term chaos here, we can say that chaotic solutions appear to be unpredictable. Small changes in initial conditions can make large differences in the solution. In this particular case, it is evidenced by the fact that ENSO occurs at irregular and apparently unpredictable intervals. It becomes even more interesting when seasonal variations in the currents are introduced.
Problem 7: Replace the equation for u' by
u' = ( B / )( T_{e} T_{w })  C( u  Au* [ 1+ sin(2 pi t) ] )
Again, approximate the solution as in Problem 6. What is the physical significance of the change in the equations? What effect does it have on the results?
The model given describes the observed qualitative features of ENSO, though not the quantitative features (such as actual sea temperatures). To make a serious effort to predict actual conditions would require a better understanding of climatic variables over the entire Pacific Ocean, and indeed, the world, as implemented in a sophisticated program on a supercomputer. The current model is valuable in that it allows us rather easily to understand the basic ingredients to a complicated physical system. It is an excellent example of a situation in which a calculation that may almost be done on the back of an envelope can provide tremendous insight into a complicated phenomenon.
References
 1
 G. K. Vallis, : A Chaotic Dynamical System?, Science, 11 April 1986.
 2
 National Center for Atmospheric Research (NCAR)
El Niño home page.
 3
 National Oceanic and Atmospheric Administration (NOAA) El Nino home page. This is an award winning source of information about this phenomenon. Images provided courtesy of the TAO Project Office of NOAA, Dr. Michael J. McPhaden, Director.
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