# Contents

## Idea

The El Niño/Southern Oscillation or ENSO is the most important form of variability in the Earth’s climate on times scales greater than a year and less than a decade. It is a quasi-periodic phenomenon that occurs across the tropical Pacific Ocean every 3 to 7 years, and on average every 4 years.

The ENSO can causes extreme weather such as floods, droughts and other weather disturbances in many regions of the world. Developing countries dependent upon agriculture and fishing, particularly those bordering the Pacific Ocean, are the most affected.

This animation produced by the Australian Bureau of Meteorology shows how the ENSO works:

During El Niño years, trade winds in the tropical Pacific weaken, and the eastern part of this ocean warms up. During La Niña years, the trade winds get stronger again, and it’s the western part of the ocean that becomes warmer. Warmer oceans create more clouds and rain.

This cycle is linked to the Southern Oscillation: an oscillation in the difference in air pressure between the eastern and western Pacific:

The top graph shows variations in the water temperature of the tropical eastern Pacific ocean: when it’s hot we have an El Niño. The bottom graph shows the air pressure in Tahiti minus the air pressure in Darwin, Australia — this called the Southern Oscillation Index, or SOI. If you stare at the graphs a while, you’ll see they’re quite strongly correlated — or more precisely, anticorrelated, since one tends to go up when the other goes down. You’ll also see that the cycles are far from perfectly periodic.

The ENSO has been changing in a number of ways in the last few decades. Some attribute these changes to global warming, but this is still uncertain.

## Changes in the ENSO

During the last several decades the number of El Niño events has increased, and the number of La Niña events has decreased. The question is whether this is a random fluctuation or a normal instance of variation for that phenomenon, or the result of global climate changes towards global warming:

• Gabriel A. Vecchi and Andrew T. Wittenberg, El Niño and our future climate: where do we stand?, WIRES Climate Change, (9 February 2010).

• Kevin E. Trenberth and Timothy J. Hoar, The 1990-1995 El Niño-Southern Oscillation event: Longest on record, Geophysical Research Letters 23 (January 1996), 57–60.

Some studies of historical data show that the recent El Niño variation may be linked to global warming. For example, one of the most recent results is that even after subtracting the positive influence of decadal variation, shown to be possibly present in the ENSO trend, the amplitude of the ENSO variability in the observed data still increases, by as much as 60% in the last 50 years:

It is not certain what exact changes will happen to ENSO in the future: different models make different predictions. It may be that the observed phenomenon of more frequent and stronger El Niño events occurs only in the initial phase of the global warming, and then (e.g., after the lower layers of the ocean get warmer as well), El Niño will become weaker than it was. It may also be that the stabilizing and destabilizing forces influencing the phenomenon will eventually compensate for each other. More research is needed to provide a better answer to that question, but the current results do not exclude the possibility of dramatic changes. See:

Abstract: The variability of ENSO, the largest interannual climate variation of the Pacific ocean-atmosphere system, and its relation to the Pacific Decadal Oscillation and global warming are documented. Analysis using the Empirical Mode Decomposition method, which is useful for analyzing nonlinear, nonstationary climate records, reveals that ENSO contains strong seasonal, biannual, decadal signals, as well as a monotonic trend that is shown to be tied closely to global warming. The frequencies of the interannual components of ENSO are higher when the decadal components of ENSO are in the warm phase, and are also increasing with the global warming trend. It is also argued that the decadal signals and the trend are connected to for the abnormal ENSO events in the 1990s.

## Causes of the ENSO

Winds called trade winds blow west across the tropical Pacific. During La Niña years, water at the ocean’s surface moves west with the wind, warming up in the sunlight as it travels. So, warm water collects at the ocean’s surface in the western Pacific. This creates more clouds and rainstorms in Asia. Meanwhile, since surface water is being dragged west by the wind, cold water from below gets pulled up to take its place in the eastern Pacific, off the coast of South America.

Furthermore, because the ocean’s surface is warmer in the western Pacific, it heats the air and makes it rise. This helps the trade winds blow west: wind blows west to fill the ‘gap’ left by rising air.

So, it’s a kind of feedback loop: the oceans being warmer in the western Pacific helps the trade winds blow west, and that makes the western oceans warmer than the eastern ones.

Of course, one may ask: why do the trade winds blow west?

Without an answer to this, the story so far would work just as well if we switched the words ‘west’ and ‘east’. That wouldn’t mean the story is wrong. It might just mean that there were two stable states of the Earth’s climate: a La Niña state where the trade winds blow west, and another state—say, the El Niño—where they blow east. One could imagine a world permanently stuck in one of these phases. Or, perhaps it could flip between these two phases for some reason.

Something roughly like the last choice is actually true. But it’s not so simple: there’s not a complete symmetry between west and east.

Why not? Mainly because the Earth is turning to the east. Air near the equator warms up and rises, so new air from more northern or southern regions moves in to take its place. But because the Earth is fatter at the equator, the equator is moving faster to the east. So, this new air from other places is moving less quickly by comparison… so as seen by someone standing on the equator, it blows west. This is an example of the Coriolis effect.

Beware: a wind that blows to the west is called an easterly. So the westward-blowing trade winds I’m talking about are called "northeasterly trades" and "southeasterly trades" on this picture. But don’t let that confuse you.

Terminology aside, the story so far should be clear. The trade winds have a good intrinsic reason to blow west, but in the La Niña phase they’re also part of a feedback loop where they make the western Pacific warmer… which in turn helps the trade winds blow west.

But then comes an El Niño. Now for some reason the westward winds weaken. This lets the built-up warm water in the western Pacific slosh back east. And with weaker westward winds, less cold water is pulled up to the surface in the east. So, the eastern Pacific warms up. This makes for more clouds and rain in the eastern Pacific—that’s when we get floods in Southern California. And with the ocean warmer in the eastern Pacific, hot air rises there, which tends to counteract the westward winds even more.

In other words, all the feedbacks reverse themselves. But note: the trade winds never mainly blow east. During the El Niño they still blow west, just a bit less. So, the climate is not flip-flopping between two symmetrical alternatives. It’s flip-flopping between two asymmetrical alternatives.

One remaining question is: why do the westward trade winds weaken? We could also ask the same question about the start of the La Niña phase: why do the westward trade winds get stronger then?

The short answer is that nobody is exactly sure. Or at least there’s no one story that everyone agrees on. There are actually several stories… and perhaps more than one of them is true. So, at this point it is worthwhile revisiting the actual data:

The top graph shows variations in the water temperature of the tropical Eastern Pacific ocean. When it’s hot we have El Niños: those are the red hills in the top graph. The blue valleys are La Niñas. Note that it’s possible to have two El Niños in a row without an intervening La Niña, or vice versa!

The bottom graph shows the Southern Oscillation Index or SOI. This is basically the air pressure in Tahiti minus the air pressure in Darwin, Australia.

So, when the SOI is high, the air pressure is higher in the east Pacific than in the west Pacific. This is what we expect in an La Niña: that’s why the westward trade winds are strong then! Conversely, the SOI is low in the El Niño phase. This variation in the SOI is called the Southern Oscillation.

If you look at the graphs above, you’ll see how one looks almost like an upside-down version of the other. So, El Niño/La Niña cycle is tightly linked to the Southern Oscillation.

Another thing you’ll see from is that ENSO cycle is far from perfectly periodic! Here’s a graph of the Southern Oscillation Index going back a lot further:

This graph was made by William Kessler. His explanations of the ENSO cycle are well worth reading:

It is worthwhile seeing his comments on theories about why an El Niño starts, and why it ends. These theories involve three additional concepts:

• The thermocline is the border between the warmer surface water in the ocean and the cold deep water, 100 to 200 meters below the surface. During the La Niña phase, warm water is blown to the western Pacific, and cold water is pulled up to the surface of the eastern Pacific. So, the thermocline becomes deeper in the west than the east. When an El Niño occurs, the thermocline flattens out:
• Oceanic Rossby waves are very low-frequency waves in the ocean’s surface and thermocline. At the ocean’s surface they are only 5 centimeters high, but hundreds of kilometers across. The surface waves are mirrored by waves in the thermocline, which are much larger, 10-50 meters in height. When the surface goes up, the thermocline goes down.

• The Madden-Julian oscillation is a pulse that moves east across the Indian Ocean and Pacific ocean at 4-8 meters/second. It manifests itself as patches of anomalously high rainfall and also anomalously low rainfall. Strong Madden-Julian Oscillations are often seen 6-12 months before an El Niño starts.

With this bit of background, we hope the reader is prepared to understand what Kessler wrote in his El Niño FAQ:

There are two main theories at present. The first is that the event is initiated by the reflection from the western boundary of the Pacific of an oceanic Rossby wave (type of low-frequency planetary wave that moves only west). The reflected wave is supposed to lower the thermocline in the west-central Pacific and thereby warm the SST sea surface temperature by reducing the efficiency of upwelling to cool the surface. Then that makes winds blow towards the (slightly) warmer water and really start the event. The nice part about this theory is that the Rossby waves can be observed for months before the reflection, which implies that El Niño is predictable.

The other idea is that the trigger is essentially random. The tropical convection (organized large-scale thunderstorm activity) in the rising air tends to occur in bursts that last for about a month, and these bursts propagate out of the Indian Ocean (known as the Madden-Julian Oscillation). Since the storms are geostrophic (rotating according to the turning of the earth, which means they rotate clockwise in the southern hemisphere and counter-clockwise in the north), storm winds on the equator always blow towards the east. If the storms are strong enough, or last long enough, then those eastward winds may be enought to start the sloshing. But specific Madden-Julian Oscillation events are not predictable much in advance (just as specific weather events are not predictable in advance), and so to the extent that this is the main element, then El Niño will not be predictable.

In my opinion both these two processes can be important in different El Niños. Some models that did not have the MJO storms were successful in predicting the events of 1986-87 and 1991-92. That suggests that the Rossby wave part was a main influence at that time. But those same models have failed to predict the events since then, and the westerlies have appeared to come from nowhere. It is also quite possible that these two general sets of ideas are incomplete, and that there are other causes entirely. The fact that we have very intermittent skill at predicting the major turns of the ENSO cycle (as opposed to the very good forecasts that can be made once an event has begun) suggests that there remain important elements that are await explanation.

## Modelling the ENSO

### The Zebiak–Cane model

Apparently the simplest climate model that exhibits somewhat realistic ENSO behavior is the ‘minimal model’ of the equatorial ocean-atmosphere system called the Zebiak–Cane model or ZC model:

• S. E. Zebiak and M. A. Cane, A model El Niño–Southern Oscillation, Mon. Weather Review 115 (1987), 2262–2278.

• J. D. Neelin, D. S. Battisti, A. C. Hirst et al., ENSO theory, J. Geophys. Res. 103(1998), 14261–14290.

For a quick overview see:

### Hopf bifurcation

In the Zebiak–Cane model, when the ocean-atmosphere coupling strength $\beta$ rises above a certain critical value ${\beta }_{c}$, a Hopf bifurcation occurs. In other words, for $\beta <{\beta }_{c}$, there is a stable equilibrium for the behavior of ocean and atmosphere, or in mathematical language, an attractive fixed point. However, for $\beta >{\beta }_{c}$ the ocean and atmosphere display periodic behavior, or in mathematical language, a limit cycle.

A picture is probably worth a thousand words, so here is a picture of a Hopf bifurcation where ${\beta }_{c}=0$:

You can see that the solution spirals in to a stable fixed point for $\beta <0$, but approaches a circle—the ‘limit cycle’—for $\beta >0$. This picture is taken from

Perhaps the simplest differential equation exhibiting a Hopf bifurcation is:

(1)$\frac{dx}{dt}=-\omega y+\left(\beta -{\beta }_{c}\right)x-x\left({x}^{2}+{y}^{2}\right)${d x\over d t} = - \omega y + (\beta - \beta_c) x - x (x^2 + y^2)
(2)$\frac{dy}{dt}=+\omega x+\left(\beta -{\beta }_{c}\right)y-y\left({x}^{2}+{y}^{2}\right)${d y\over d t} = + \omega x + (\beta - \beta_c) y - y (x^2 + y^2)

This is much clearer in polar coordinates:

(3)$\frac{dr}{dt}=\left(\beta -{\beta }_{c}\right)r-{r}^{3}${d r\over d t} = (\beta - \beta_c)r - r^3
(4)$\frac{d\theta }{dt}=\omega${d \theta\over d t} = \omega

These say that the solution goes round and round at constant angular velocity $\omega$, while the distance from the origin, $r$, approaches either $0$ if $\beta <{\beta }_{c}$ or the unique positive solution of

$\left(\beta -{\beta }_{c}\right)r-{r}^{3}=0$(\beta - \beta_c)r - r^3 = 0

if $\mathrm{beta}>{\beta }_{c}$. Solving this equation, we see in the latter case that we get a limit cycle with radius

$r=\sqrt{\beta -{\beta }_{c}}\phantom{\rule{thinmathspace}{0ex}}.$r = \sqrt{\beta - \beta_c}\, .

For more, see Hopf bifurcation.

### Delayed action oscillator

A rather different pedagogical toy model of ENSO may be found here:

In its simplest form, this model uses a delay-differential equation:

$\frac{d}{dt}T\left(t\right)=kT\left(t\right)-bT\left(t{\right)}^{3}-AT\left(t-\Delta \right)${d\over d t} T(t) = k T(t) - b T(t)^3 - A T(t - \Delta)

($k,b,A,\Delta >0$) to model the possible effect of oceanic Rossby waves. By rescaling time appropriately and redefining the constants we get the dimensionless form, equation (2) of the paper:

$\frac{d}{dt}T\left(t\right)=T\left(t\right)-T\left(t{\right)}^{3}-AT\left(t-\Delta \right)${d\over d t} T(t) = T(t) - T(t)^3 - A T(t - \Delta)

Quoting the paper:

Lastly, the model also considers equatorially-trapped ocean waves propagating across the Pacific, before interacting back with the central Pacific region after a certain time delay. These ocean waves are “hidden” Rossby waves which move westward on the thermocline, reflect off the rigid continental boundary in the West and then return eastward along the equator as Kelvin waves.

The delay term has a negative coefficient representing a negative feedback. To see the reason for this, let us consider a warm SST sea surface temperature perturbation in the coupled region. This produces a westerly wind response that deepens the thermocline locally (immediate positive feedback), but at the same time, the wind perturbations produce divergent westward propagating Rossby waves that return after time $\Delta$ to create upwelling and cooling, reducing the original perturbation.

Note that equation (9) from this paper,

$\frac{d}{dt}T\left(t\right)=T\left(t\right)-T\left(t{\right)}^{3}-AT\left(t-\Delta \right)+\frac{d}{dt}Y\left(t\right)${d\over d t} T(t) = T(t) - T(t)^3 - A T(t - \Delta) + \frac{d}{d t} Y(t)

with an annual forcing resembles the bistable potential on the page stochastic resonance in the case of a periodic forcing, with an additional linear time-delayed feedback term.

### Stochastic aspects

The Hopf bifurcation model above gives perfectly periodic behavior, so it does not explain the irregularity of the ENSO cycle. The peculiar variability in the ‘period’ of the ENSO has been studied with the help of stochastic differential equations:

Abstract: The El Niño/Southern Oscillation (ENSO) phenomenon is the dominant climatic fluctuation on interannual time scales. It is an irregular oscillation with a distinctive broadband spectrum. In this article, we discuss recent theories that seek to explain this irregularity. Particular attention is paid to explanations that involve the stochastic forcing of the slow ocean modes by fast atmospheric transients. We present a theoretical framework for analysing this picture of the irregularity and also discuss the results from a number of coupled ocean–atmosphere models. Finally, we briefly review the implications of the various explanations of ENSO irregularity to attempts to predict this economically significant phenomenon.

Kleeman actually discusses two general theories for the irregularity of the ENSO:

1. Perhaps the slowly varying modes involved interact with each other in a chaotic way.

2. Perhaps the slowly varying modes involved interact with each other in a non-chaotic way, but also interact with rapidly-varying chaotic modes, which inject noise into what would otherwise be simple periodic behavior.

Of course there is a third option: Perhaps the slowly varying modes involved interact with each other in a chaotic way, but also interact with rapidly-varying chaotic modes. However, Kleeman concentrates on the second option.

As a toy model, Kleeman describes the behavior of version of equations (1) and (2) to which white noise has been added. This produces irregular cyclic behavior even for $\beta <{\beta }_{c}$. Roughly speaking, the noise keeps knocking the solution away from the stable fixed point at $x=y=0$, so it keeps going round and round, but in an irregular way.

Note that a transformation from Cartesian coordinates to polar coordinates in the presence of noise cannot use the usual chain rule of calculus, but has to use e.g. the Itô formula for Itô stochastic differential equations instead.

For a talk on this subject, see:

It is also interesting to consider a stochastic version of the delayed action oscillator. For details, see

and also our page on stochastic delay differential equations?.

## References

As usual, a good place to start is the Wikipedia article:

Also see:

You can download the SOI data from the National Oceanic and Atmospheric Administration (NOAA) website, and also read how the SOI is computed.

category: climate