We consider a delay differential equation (DDE) model for El-Nino Southern Oscillation (ENSO) variability. The model combines two key mechanisms that participate in ENSO dynamics: delayed negative feedback and seasonal forcing. We perform stability analyses of the model in the three-dimensional space of its physically relevant parameters. Our results illustrate the role of these three parameters: strength of seasonal forcing $b$, atmosphere-ocean coupling $\kappa$, and propagation period $\tau$ of oceanic waves across the Tropical Pacific. Two regimes of variability, stable and unstable, are separated by a sharp neutral curve in the $(b,\tau)$ plane at constant $\kappa$. The detailed structure of the neutral curve becomes very irregular and possibly fractal, while individual trajectories within the unstable region become highly complex and possibly chaotic, as the atmosphere-ocean coupling $\kappa$ increases. In the unstable regime, spontaneous transitions occur in the mean ``temperature'' ({\it i.e.}, thermocline depth), period, and extreme annual values, for purely periodic, seasonal forcing. The model reproduces the Devil's bleachers characterizing other ENSO models, such as nonlinear, coupled systems of partial differential equations; some of the features of this behavior have been documented in general circulation models, as well as in observations. We expect, therefore, similar behavior in much more detailed and realistic models, where it is harder to describe its causes as completely.
Deep Dive into A delay differential model of ENSO variability: Parametric instability and the distribution of extremes.
We consider a delay differential equation (DDE) model for El-Nino Southern Oscillation (ENSO) variability. The model combines two key mechanisms that participate in ENSO dynamics: delayed negative feedback and seasonal forcing. We perform stability analyses of the model in the three-dimensional space of its physically relevant parameters. Our results illustrate the role of these three parameters: strength of seasonal forcing $b$, atmosphere-ocean coupling $\kappa$, and propagation period $\tau$ of oceanic waves across the Tropical Pacific. Two regimes of variability, stable and unstable, are separated by a sharp neutral curve in the $(b,\tau)$ plane at constant $\kappa$. The detailed structure of the neutral curve becomes very irregular and possibly fractal, while individual trajectories within the unstable region become highly complex and possibly chaotic, as the atmosphere-ocean coupling $\kappa$ increases. In the unstable regime, spontaneous transitions occur in the mean ``temperatu
The following conceptual elements have been shown to play a determining role in the dynamics of the ENSO phenomenon.
(i) The Bjerknes hypothesis: Bjerknes (1969), who laid the foundation of modern ENSO research, suggested a positive feedback as a mechanism for the growth of an internal instability that could produce large positive anomalies of sea surface temperatures (SSTs) in the eastern Tropical Pacific.
We use here the climatological meaning of the term anomaly, i.e., the difference between an instantaneous (or short-term average) value and the normal (or long-term mean). Using observations from the International Geophysical Year (1957)(1958), Bjerknes realized that this mechanism must involve air-sea interaction in the tropics. The “chain reaction” starts with an initial warming of SSTs in the “cold tongue” that occupies the eastern part of the equatorial Pacific. This warming causes a weakening of the thermally direct Walkercell circulation; this circulation involves air rising over the warmer SSTs near Indonesia and sinking over the colder SSTs near Peru. As the trade winds blowing from the east weaken and thus give way to westerly wind anomalies, the ensuing local changes in the ocean circulation encourage further SST increase. Thus the feedback loop is closed and further amplification of the instability is triggered.
(ii) Delayed oceanic wave adjustments: Compensating for Bjerknes’s positive feedback is a negative feedback in the system that allows a return to colder conditions in the basin’s eastern part (Suarez and Schopf, 1988). During the peak of the cold-tongue warming, called the warm or El Niño phase of ENSO, westerly wind anomalies prevail in the central part of the basin. As part of the ocean’s adjustment to this atmospheric forcing, a Kelvin wave is set up in the tropical wave guide and carries a warming signal eastward; this signal deepens the eastern-basin thermocline, which separates the warmer, well-mixed surface waters from the colder waters below, and thus contributes to the positive feedback described above. Concurrently, slower Rossby waves prop-agate westward, and are reflected at the basin’s western boundary, giving rise therewith to an eastward-propagating Kelvin wave that has a cooling, thermocline-shoaling effect.
Over time, the arrival of this signal erodes the warm event, ultimately causing a switch to a cold, La Niña phase.
A growing body of work (Ghil and Robertson, 2000;Chang et al., 1994Chang et al., , 1995;;Jin et al., 1994Jin et al., , 1996;;Tziperman et al., 1994Tziperman et al., , 1995) ) points to resonances between the Pacific basin’s intrinsic air-sea oscillator and the annual cycle as a possible cause for the tendency of warm events to peak in boreal winter, as well as for ENSO’s intriguing mix of temporal regularities and irregularities. The mechanisms by which this interaction takes place are numerous and intricate and their relative importance is not yet fully understood (Tziperman et al., 1995;Battisti, 1988).
Starting in the 1980s, the effects of delayed feedbacks and external forcing have been studied using the formalism of delay differential equations (DDE) (see, inter alia, Bhattacharrya and Ghil (1982); Ghil and Childress (1987) for geoscience applications, and Hale (1977); Nussbaum (1998) for DDE theory). Several DDE systems have been suggested as toy models for ENSO variability. Battisti and Hirst (1989) have considered the linear autonomous DDE dT /dt = -α T (t -τ ) + T, α > 0, τ > 0.
(1)
Here, T represents the sea-surface temperature (SST) averaged over the eastern equatorial Pacific. The first term on the right-hand side (rhs) of (1) mimics the negative feedback due to the oceanic waves, while the second term reflects Bjerknes’s positive feedback. As shown in (Battisti and Hirst, 1989), Eq. ( 1) reproduces some of the main features of a fully nonlinear coupled atmosphere-ocean model of ENSO dynamics in the tropics (Battisti, 1988;Zebiak and Cane, 1987). Suarez and Schopf (1988) and Battisti and Hirst (1989) also studied a nonlinear version of (1), in which a cubic nonlinearity is added to the rhs of the equation:
where 0 < α < 1 and τ > 0. This system has three steady states, obtained by finding the roots of the rhs:
The so-called inner solution T 0 is always unstable, while the outer solutions T 1,2 may be stable or unstable depending on the parameters (α, τ ). If an outer steady state is unstable, the system exhibits bounded oscillatory dynamics; in (Suarez and Schopf, 1988) it was shown numerically that a typical period of such oscillatory solutions is about two times the delay τ .
The delay equation idea was very successful in explaining the periodic nature of ENSO events. Indeed, the delayed negative feedback does not let a solution fade away or blow up, as in the ordinary differential equation (ODE) case with τ = 0, and thus creates an internal oscillator with period depending on the delay and particular form of the equation’s rhs.
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