1. ENSO and its relation to the Kelvin and Rossby Wave Eric Sinsky

2. Outline Basic El Nino and La Nina dynamics
The basic principles of the Kelvin Wave
The basic principles of the Rossby Wave
The Kelvin wave’s role in ENSO
The Rossby wave’s role in ENSO
An ENSO theory involving Rossby and Kelvin waves

3. El Nino
Figure (NOAA)
In normal conditions, the ocean currents are moving from east to west at the equator. The winds are also moving westward at the equator. The high pressure in the eastern boundary implies a lot of days with clear skies. This allows a lot of shortwave radiation to contact the sea surface. The currents move the warm water westward creating a warm pool. So the sea surface temperature is warmer in the west and cooler in the east. The west boundary has a lower pressure than the east. This causes the air to rise creating storm systems.
In el Nino conditions, the trade winds weaken. This causes the pressure gradient force to dominant the influence of wind stress, resulting in the currents moving from west to east. The wind currents at the east boundary move east to west and rise in the mid pacfic around Tahiti. The wind currents in the west boundary move west to east and also rise in the mid pacific around Tahiti. The point where the winds meets is a lot pressure system, where an anonymously high amount of precipitation occurs. The west boundary is known for having anonymously dry weather and the east boundary having anonymously warm sea surface temperatures during El Nino. Also the thermocline is less steep than average.

4. La Nina
Figure (NOAA)
During La Nina, there is a strengthening of the trade winds. Sea surface temperature is anonymously cool in the eastern boundary. The thermocline is steeper than average. The winds and currents move at the same direction as normal conditions.

5. How ENSO is quantified ENSO is measured in SOI
Positive values signify La Nina and negative El Nino
figure – Stewart
When Sea surface temperature anomalies are above 0.5C in the El Nino 3.4 region (170E-120W), it is considered ENSO positive (warm) phase (El Nino).
When Sea surface temperature anomalies are below -0.5C in the El Nino 3.4 region (170E-120W), it is considered ENSO negative (cool) phase (La Nina).
El Nino can be measured in a few ways.
The SOI index is calculated by finding the [(sea level pressure difference – the long term average of the sea level pressure difference) /(Standard deviation of the long term of the sea level pressure difference)]
The Standardized SOI index is calculated by finding the [(sea level pressure anomaly of Tahiti and divide it by its standard deviation) – (sea level pressure anomaly of Darwin divided by its standard deviation)]/(standard deviation of the difference). The means are found from
The figure shown is a normalized Southern Oscillation line plot.
Notable El Nino years include and
On average, an El Nino and La Nina happens every 2-7 years and they last for about

6. El Nino
Dark red represents high sea surface temperature anomalies and dark blue represents cool SST anomalies.
Since the SST anomalies are above 0.5C in the El Nino 3.4 region, this is considered an El Nino.

7. Thermocline Depth
Notice the temperature becomes more correlated with the thermocline depth the further east you go along the equatorial Pacific. The east Pacific is where the thermocline deepens the most because of the famous El Nino.

8. Kelvin Wave Shallow water wave ( ℎ λ < 1 20 )
Small amplitude (5-10cm)
Very long wavelength and period
Non-dispersive
C= 𝐶 𝑔 = 𝑔′ℎ , η(𝑡,𝑥,𝑦)= η 0 cos (κ 𝑦−𝜔𝑡) 𝑒 −𝑓𝑥 𝐶
Kelvin waves are usually 5-10cm in amplitude (“A Curious Pacific Wave”)
Non-dispersive (phase speed = group speed) ; typically 1-2 m/s
Figure (NPS)
Waves are a result of an abrupt change in wind velocity. (NPS)

9. Rossby Wave Deep water wave ( ℎ λ > 1 4 )
Large amplitude at thermocline (~25m)
Very long wavelength (500km)
Dispersive
𝑐= − 𝜕𝑓 𝜕𝑦 κ 2 + 𝑓 2 𝑔ℎ
Figure – National Oceanography Centre (University of Southampton)
Rossby wave exist along the thermocline
They move about 10cm/s

10. Kelvin Wave in ENSO Figure (A curious Pacific Wave)
Notice the Kelvin wave propagates eastward. During this El Nino, the warm pool expands eastward as shown in the figure.

11. Rossby Waves in Enso
Figure (Chelton, D. B., and M. G. Schlax, 1996: Global observations of oceanic Rossby waves. Science, vol. 272, pp )
We will use the 1993 El Nino as an example to show the rossby wave in a spatial structure. The white solid line in the left figure represents the trough of the westward propagating rossby wave. Kelvin wave trough is the “X” and the Rossby wave crest is the open circle and triangle.
Solid circles are Pacific data and open circles are Indian and Atlantic data.
Notice the observed phase speed tends to be more than the theoretical, especially at high latitudes.
Below there is a plot that shows the ratio of observed to theoretical phase speed.

12. Comparing Rossby and Kelvin Waves
Figure – Stewart
Kelvin wave is on the right propagating eastward and the Rossby wave is on the left propagating westward.

13. The Delayed Oscillator
The delayed oscillator is a famous ENSO theory that involves an instability in the equatorial Pacific followed by a stabilization process through the Kelvin and Rossby wave. The theory was started by Battisti and Hirst (1989), and Suarez and Schopf (1988).
First, positive SST anomalies in the east boundary causes wind stress anomalies. The wind stress anomalies result in a weakening of the trade winds.
Second, the weakening of the trade winds causes a westward propagating upwelling rossby wave which shallow the thermocline in the mid to west Pacific, which in turn makes even higher SST anomalies. Also, an eastward propagating downwelling kelvin wave is formed which deepens the thermocline in the east. This results in a less steep thermocline and a weakening in the coastal upwelling. This is the point where El Nino is at its peak.
Third, as soon as the Kelvin Wave reflects off the east boundary, it turns into a westward propagating upwelling Rossby wave. This westward propagating upwelling rossby wave is the fastest sort of rossby wave. This rossby wave forms a shallower thermocline to the west.
As soon as the Rossby wave reflects off the west boundary, it turns into an eastward propagating upwelling Kelvin wave, the thermocline is raised in the east, the SST anomalies approach 0 and even becomes negative (SST cool down) and El Nino terminates. When the Kelvin wave is formed, it is responsible for the delayed response for negative SST anomalies. Overall, El Nino has “seeds” for its own destruction.
Sometimes the SST anomaly becomes highly negative resulting in a La Nina.
The time frame of the westward propagation of the rossby wave is 12 to 18 months.
Kelvin waves cross the pacific in only 3 months.

14. Delayed oscillator applied to 97-98 El Nino
This is an example of the delayed oscillator theory in the El Nino. As discussed earlier, the isotherm depth is deepening in the east due to the eastward propagating kelvin wave as observed in the figure. Also, the isotherm depth is becoming more shallow in the west due to the westward propagating rossby wave. One noticeable aspect in this figure is we know when the Rossby wave reflects off the west boundary. It happens right before SST anomalies cool in the east.

15. References
Chelton, D. B., and M. G. Schlax, 1996: Global observations of oceanic Rossby waves. Science, vol. 272, pp
Knauss, John. Introduction of Physical Oceanography. Long Grove: Waveland Press, 1997.
Stewart, Robert. Introduction to Physical Oceanography. Texas A&M University, 2005.
Cipollini, Paolo. Rossby Waves: what are they? University of Southamton, 2000.
Anonymous. A Curious Pacific Wave. National Aeronautics and Space Administration Science, 2002.
Dijkstra, Henk. Nonlinear Physical Oceanography. New York: Springer-Verlag, 2005.
Laing, Arlene, and Evans, Jenni-Louise. Introduction to Tropical Meteorology. COMET® Program, 2011.
 Harrison, D. E. and G. A. Vecchi (2001), El Niño and La Niña—equatorial Pacific thermocline depth and sea surface temperature anomalies, 1986–98, Geophys. Res. Lett., 28(6), 1051–1054, doi: /1999GL