The Azimuth Project
Experiments in ENSO modeling (Rev #4)

ENSO is the standing wave mode of the multi-year Equatorial Pacific waters.


The introductory background is here

The sloshing motion of the volume suggests that we can use conventional hydrodynamics models of volumes of water to characterize the past and potentially future ENSO behavior.

The basic idea is described in this white paper called “Sloshing Model for ENSO”

As this sloshing model evolves, this page will serve as a place to present updates.

The sloshing model is essentially a second-order wave equation with a modulation of the characteristic frequency ala the Hill or Mathieu equation.

f(t)+(a2qcos2t)f(t)=F(t) f''(t) + (a - 2q \cos{2t}) f(t) = F(t)

The term a is ω 2\omega^2, the characteristic frequency of the resonant wave while the q term defines the amplitude of the Mathieu modulation. The former is well accepted in research circles (see for example work by Allan Clarke), while the latter is an important perturbation well known only in the hydrodynamics of sloshing for smaller volumes (see recent groundbreaking work by Frandsen and Faltinsen). What separates the Mathieu equation from the Hill equation is that Hill allows a generalized set of sinusoidal terms instead of the single sinusoid in the Mathieu formulation.

f(t)+Hill(t)*f(t)=F(t) f''(t) + Hill(t)*f(t) = F(t)

The right-hand side (RHS) term, F(t) is the forcing applied to the sloshing, which is the stimulus needed to sustain the quasi-periodic oscillations in a standing-wave mode.

The challenge is to come up with physical models for the parameters and forcing.

The characteristic frequency is set to a period of 4.25 years following Clarke. This becomes the constant term, a, in the Hill equation.

Through experimentation, the Hill modulation term appears to follow closely the Total Solar Irradiation (TSI) variation observed from solar measurements.

The forcing involves angular momentum changes characterized by Chandler wobble (CW) measurements as well as the measurements of the upper atmosphere quasi-biennial oscillations (QBO) in wind speed and directions. Together, these provide the sea-saw stimulation that drives the ENSO sloshing.

To solve the general equation, we use Mathematica to do the integration of the tricky non-linear formulation. The best way to do this is to first emulate the measured time-series data as a Fourier series of sinusoidal terms and optimize the correlation of model with data.

To start, we model the TSI data:


This has a correlation coefficient of over 0.96 and this level is necessary as there is a strong long term modulation which obscures the Hale cycle of 20 years and less.

Next we model the QBO data at 20 hPa:


This has some jitter in the fundamental period of 2.33 years, but only goes back to 1953.

Finally the Chandler wobble data:


Putting these components together in the sloshing equation and adjusting the scale factors to match the ENSO SOI time-series data, we arrive at this correlation:


To improve the fit slightly, two evident but secondary periodic components of 9.08 years and 7.3 years were added to the result. The term 9.08 years is very close to the average of the diurnal and semi-diurnal long term tidal periods of 9.3 and 8.85 years. While the 7.3 year period is a cycle that is identified from the global temperature signal.