1. Introduction

El Niño–Southern Oscillation (ENSO) ocean–atmosphere coupling is usually most intense in the equatorial Pacific Ocean between about 160°E and 160°W, for it is here that the surface ENSO winds are strong and zonal and where the anomalous equatorial atmospheric convection is greatest (see Fig. 1). Many ENSO studies have assumed that ENSO coupling is strong in the eastern equatorial Pacific, but Fig. 1 shows that there the zonal band of anomalous precipitation north of the equator drives mainly meridional winds with weak wind stress curl. Such wind forcing does not generate significant upwelling or ocean flow and consequently ocean–atmosphere coupling there is weak.

Large variability in deep atmospheric convection is centered on the data line because the date line marks the edge of the western Pacific warm pool and the large-scale deep convection lying above it; zonal displacements of the warm pool edge cause zonal displacements of deep atmospheric convection and precipitation (see Fig. 2 as well as Fu et al. 1986) and large anomalies of deep atmospheric convection result (Fig. 1). Such deep anomalous equatorial convection produces anomalous surface equatorial westerlies (Fig. 1). These westerly wind anomalies drive eastward flow that advects the warm pool eastward (Gill 1983; McPhaden and Picaut 1990; Picaut and Delcroix 1995; Picaut et al. 1996), resulting in further anomalous convection, stronger anomalous westerlies, further eastward warm pool displacement, etc. (Gill and Rasmusson 1983; Picaut et al. 1997). This unstable growth is arrested by negative feedbacks, the most important of these dynamically being the delayed negative feedback associated with long ocean waves generated by the anomalous westerly winds. For example, the westerly winds generate long equatorial Rossby waves that propagate to the western Pacific boundary, reflect as equatorial Kelvin waves, and, after the appropriate wave propagation delay, reach the edge of the warm pool. The equatorial Kelvin waves are associated with a westward flow tending to return the warm pool to its original position; that is, there is delayed negative feedback. Theoretical and observational evidence for such western boundary reflection has been provided by Clarke (1991), du Penhoat and Cane (1991), Li and Clarke (1994), Mantua and Battisti (1994), and Boulanger and Fu (1996). Evidence for the influence of these long waves on the zonal equatorial flow and displacement of the warm pool has been given by Gill (1983), Picaut and Delcroix (1995), and Picaut et al. (1996).

Delayed negative feedback may also occur because of ocean wave reflection at the eastern ocean boundary. Equatorial Kelvin waves directly generated by the instability propagate eastward to the eastern ocean boundary and reflect as equatorial Rossby waves with westward equatorial currents. These currents act to advect the warm pool westward towards its undisturbed position. Evidence for the influence of eastern boundary wave reflection on zonal current anomalies has been provided by Delcroix et al. (1994), Picaut and Delcroix (1995), Boulanger and Fu (1996), and Picaut et al. (1997).

The mean zonal flow converges at the eastern edge of the warm pool (Picaut et al. 1996) and this also tends to brake the growing warm pool displacement, tending to restore it to the mean position. Other negative feedbacks also operate and these will be discussed briefly in section 3a(2).

As pointed out by Picaut et al. (1997), one rare complication with the warm pool displacement model occurs for the very strong El Niño case (like 1982–83 and 1997–98) when the edge of the warm pool (e.g., the 28.5°C isotherm) reaches the eastern boundary and dives beneath the ocean surface. In this case upwelling as well as zonal advection are important to the dynamics. However, the delayed negative feedback through long Kelvin and Rossby waves still operates. In fact, in section 3 we will show that the warm pool displacement model equations have a very similar structure to Battisti and Hirst’s (1989) delayed oscillator ENSO model for which upwelling in the eastern equatorial Pacific and delayed negative feedback associated with long Kelvin and Rossby waves are key components.

Picaut et al. (1997) used a numerical model to show that the combination of growing displacement, delayed negative long equatorial wave feedback, and mean zonal flow gives rise to an interannual oscillation with periodicity comparable to that in ENSO. They pointed out that this warm pool displacement theory of ENSO is different physically from the delayed oscillator theory of ENSO (Schopf and Suarez 1988; Battisti 1988; Suarez and Schopf 1988; Battisti and Hirst 1989). We will show that, although the two theories differ physically, they do have a similar mathematical form. The main difference mathematically is that the warm pool displacement model has important nonlinearity, namely, that the wave propagation delay time varies as the distance of the edge of the warm pool from the ocean boundaries varies.

In section 2 we formulate a simple warm pool displacement ENSO theory and then obtain some analytical and numerical solutions in section 3. Section 4 shows that, because the wave transit time depends on the location of the eastern edge of the warm pool, the model time series has a structure similar to that observed in that the bigger the warm ENSO episode, the longer to the next cold ENSO episode. A final section summarizes the main results.

2. Model formulation

The anomalous zonal ocean current u′ consists of three parts, which we denote u^′₁, u^′₂, and u^′₃. When x is positive, the warm pool will be eastward of its normal position and anomalous deep atmospheric convection will generate westerly surface equatorial winds [see Fig. 1 and the physical explanation of Clarke (1994)]. These anomalous eastward winds generate an anomalous eastward local flow u^′₁ and two delayed negative feedback flows u^′₂ and u^′₃. The local flow is essentially in phase with the anomalous wind stress and x(t) and increases as the anomalous wind stress and x(t) increase. We write

u^′₁taxt(2.3)

The second current anomaly u^′₂ results from wave propagation. The anomalous westerly winds generate Rossby waves that propagate westward, reflect at the western ocean boundary, and return as equatorial Kelvin waves even though the western Pacific equatorial boundary is irregular and gappy (Clarke 1991; du Penhoat and Cane 1991; Li and Clarke 1994; Mantua and Battisti 1994). The Rossby waves have eastward currents and the reflected equatorial Kelvin waves have westward currents in order to satisfy the western equatorial boundary condition. We therefore write

u^′₂bxt(2.4)

where Δ is the time it takes for Rossby waves to propagate from the region of strong ocean–atmosphere coupling (near the date line) to the western boundary, reflect as an equatorial Kelvin wave, and return to the region of strong ocean–atmosphere coupling. Both observations and theory (Li and Clarke 1994) suggest that it should take Rossby waves about 3 months to get to the western boundary and, theoretically, it should take the equatorial Kelvin waves 1 month to get back so Δ = 4 months. But when the warm pool moves, these travel times will change. Based on the result that the first vertical mode Kelvin wave (speed c) and the first vertical mode, first meridional mode Rossby wave (speed c/3) dominate the flow averaged over 4°S–4°N (Picaut and Delcroix 1995), we write

View Expanded

Note that (2.5) defines a nonlinear problem for Δ. It can be rapidly solved by iteration, beginning with Δ = 4 months in the expression on the right-hand side. In our calculations we took c = 2.5 m s⁻¹.

The third contribution to the current anomaly, u^′₃, is of similar form to the second. The second contribution involves generation of Rossby waves and reflection of these from the western Pacific Ocean boundary, while the third contribution involves the generation of equatorial Kelvin waves and their reflection from the eastern Pacific boundary as Rossby waves. If the anomalous winds are westerly (easterly) then the equatorial Kelvin waves have eastward (westward) currents and the reflected Rossby waves westward (eastward) ones, so

u^′₃(t) = −dx(t − δ),(2.6)

where

View Expanded

In (2.7) the average delay is 8 months because the average round trip travel time from the region of strong ocean–atmosphere coupling in the central-western equatorial Pacific to the eastern boundary is about double that to the western boundary (average Δ = 4 months). Substitution of (2.3), (2.4), and (2.6) into (2.2) gives the nonlinear equation

dxdtuxtaxtbxtdxtδ(2.8)

3. Solutions

We first discuss solutions of (2.8) when there is delayed negative feedback due to reflection from the western Pacific boundary alone, and then consider the more general case when there is delayed negative feedback from both eastern and western Pacific boundaries.

a. Model with western boundary reflection alone

1) Analytical solutions

When there is no eastern boundary reflection, d = 0 in (2.8). Initially we further simplify the problem by assuming there is no mean restoring flow (u ≡ 0) and the delay Δ is constant. Then (2.8) reduces to

dxdtaxtbxt(3.1)

This has exactly the same form as the delayed oscillator model of Battisti and Hirst (1989) and, except for a cubic damping term, has the same form as the delayed oscillator model of Schopf (1987) and Suarez and Schopf (1988). These delayed oscillator models differ physically from ours in that they are not based on warm pool displacement as the main component of ENSO physics.

Following Battisti and Hirst (1989), look for solutions of the linear equation (3.1) in the form

xtAσtωt(3.2)

by substituting A exp(σt + iωt) for x(t) in (3.1). We obtain

σiωabσiω(3.3)

Battisti and Hirst plotted and discussed solutions for σ and ω over a wide parameter range. Here our interest is in the low frequencies relevant to ENSO, and we obtain analytical results for

ζω(3.4)

To do this, multiply (3.3) by Δ and, after separating into real and imaginary parts, obtain

View Expanded

where the two dimensionless unknowns are ζ [see (3.4)] and

ξσ(3.7)

From (3.6)

be^−ξζζ,(3.8)

so

ξbζζ(3.9)

Substituting these results into (3.5) gives an equation for ζ:

bζζaζζζ(3.10)

For ζ ≲ 1, with small error

View Expanded

Thus, from (3.10), (3.11), and (3.12) we obtain

ζ2ba^1/2(3.13)

and hence, from (3.9), (3.11), and (3.13) that

View Expanded

It follows from (3.13) and ζ = ωΔ that the period T of the oscillation is

T2πba^1/2(3.15)

and from (3.14) and ξ = σΔ that the e-folding growth rate timescale σ⁻¹ is

σ⁻¹ba(3.16)

Numerical solutions of (3.1) show that the approximate analytical formulas, valid for small ζ², have small error even when ζ = 1.

The formulas (3.15) and (3.16) indicate that the model oscillations can be of ENSO periodicity for reasonable parameter values. To show this, we must first estimate a and b. From (2.4) the equatorial Kelvin waves have currents

bx⁻¹

where the typical 0.1 m s⁻¹ amplitude estimate follows from typical interannual sea level amplitudes on Australia’s western coast and theory for equatorial wave reflection at the gappy western Pacific boundary (Clarke 1991). Since typical amplitudes for x are ∼2000 km, for Δ = 4 months

b

But the Rossby wave mass flux onto the western boundary, generated remotely through the ENSO air–sea interaction, is comparable to the reflected equatorial Kelvin wave mass flux (du Penhoat and Cane 1991; Clarke 1991). Therefore, we have

axbx,

and hence

ba

For such a parameter range, (3.16) and (3.15) show that growing, neutral, or decaying long period solutions are possible.

Notice that the period T of the coupled oscillation [see (3.15)] depends both on the wave transit time Δ and the timescales a⁻¹ and b⁻¹ associated with the anomalous advecting flow. Physically, if the negative feedback is sufficiently delayed, there is time for the instability to grow before being damped out by the negative feedback. There is also time, once the damping has overpowered the instability, for the negative feedback to change the sign of x(t), for example, from positive to negative. This small negative perturbation can then grow and be restrained by delayed negative feedback so that eventually a small positive (eastward) perturbation of the warm pool results. In this way a sequence of warm and cold ENSO episodes occurs. The oscillation periodicity, which is about 2–7 years, is much longer than the 4-month wave-transit delay time; although the delay time is crucial to the existence of the period of the oscillation, the oscillation period also depends critically on the relative importance of the growth rate of the instability and the negative feedback.

2) The influence of mean flow u and other negative feedbacks

So far the mean flow u has not been considered. As mentioned earlier, it acts as a negative feedback, opposing any displacement of the mean position of the 28.5°C isotherm. For small x, by Taylor series expansion and vanishing u at the point of convergence x = 0, u(x(t)) is linear in x. Mathematically, if u is linear in x, then its inclusion in (3.1) effectively decreases a and results in a smaller growth rate [see (3.14)]. For large x, u does not change linearly (see Fig. 4) and the negative feedback is not as strong. Picaut et al. (1997) had to make the size of u in their model larger than that observed for their model to give reasonable solutions. This is because in the real world there are other negative feedbacks.

One negative feedback is associated with the easterly (westerly) wind anomalies which form in the western equatorial Pacific during warm (cold) ENSO episodes, particularly during December–February (see Fig. 5 and Weisberg and Wang 1997). These western equatorial Pacific wind anomalies generate equatorial Kelvin waves with zonal flow opposite to the flow of the growing instability near the date line. Since the equatorial Kelvin waves only take about 1 month to propagate from the western equatorial Pacific to the date line, this negative feedback is of the form −αx[t − 1 − x(t)/c] where α is a positive constant and time is in months. This term could be included in the right-hand side of (2.8). For the simplest case when it provides the only delayed negative feed back, the model equation is of the form (3.1) with Δ replaced by [1 + x(t)/c]. The Δ = const analysis of section 3a(1) still applies, so oscillations are still possible, but since the delay is so small, values of a and b have to be very finely tuned to give ENSO frequency solutions. However, in reality at least one of the other delayed negative feedback terms in (2.8) should not be ignored. Because the delay [1 + x(t)/c] months is small compared with the delays Δ [see (2.5)] or δ [see (2.7)] and because t − 1 − x(t)/c is close to t for interannual oscillations, the main effect of the term −αx[t − 1 − x(t)/c] will be to reduce the parameter a by α in the growth term ax(t).

Another negative feedback is due to the surface heat flux. Although, as mentioned earlier, Liu et al. (1994) showed that in the central equatorial Pacific anomalous net solar surface heat flux and anomalous evaporation are of minor importance to SST change, these effects may also contribute to the reduction of a.

In terms of our simple model, overall the negative feedbacks result in oscillations that do not grow or decay. For this σ = 0 case, by (3.14), our simple model has

ab(3.17)

and hence, by (3.15), the period T of the oscillation is

T2πb^1/2(3.18)

Note that (3.18) implies that low frequency periodicity is restricted to a very small range in the parameter bΔ. For Δ = 4 months, by (3.18)

Tb^1/2(3.19)

Therefore, for interannual periodicity and zero growth rate, lnbΔ must be a small number; that is, bΔ must be restricted to a number slightly greater than 1. It follows from (3.17) that aΔ must also be nearly 1 for neutrally stable oscillations. This makes sense physically. If aΔ were much greater than 1, x(t) in (3.1) would grow too much before the negative feedback could control it and produce a stable oscillation; if aΔ were small, x(t) would take much longer than Δ to grow or decay, so Δ would be negligible in (3.1) and decaying (b > a) or growing (a > b) solutions would result.

3) Nonconstant Δ

So far the delay Δ has been taken to be constant rather than the more accurate time-varying x-dependent value in (2.5). Figure 6 shows analytical and numerical solutions of (3.1) for Δ constant and Δ given by (2.5) for the reasonable parameter values given in the figure. The parameters a and b for the constant Δ case were chosen so that there was no growth in the solution. For these same parameters, Fig. 6 shows that when Δ depends on x(t), the solution is more stable, the amplitude decaying slowly in time. The amplitude decays more slowly as the amplitude decreases because a smaller amplitude means that Δ is closer to the constant Δ and σ = 0 case. If we increase the growth parameter “a” slightly, then the slow decay of the x-dependent Δ case can be removed and a constant amplitude oscillation results.

Figure 6 suggests that variable Δ does not affect the solution much. However, in section 4, when we consider irregular oscillations in which the amplitude of the warm event varies considerably for different warm event half cycles, variable Δ has a substantial effect.

4. Timing/amplitude properties in ENSO time series

The solutions obtained so far have been sinusoidal or very nearly so. Observed x(t) is irregular, and, like the Tahiti minus Darwin Southern Oscillation index (SOI), has a curious timing/amplitude property (Clarke and Li 1995). Specifically, there is a tendency that the larger the warm ENSO episode, the longer to the next cold ENSO episode. Figures 7 and 8 illustrate this property for x(t). We low-pass filtered x(t) to remove noise so that the extrema of x(t) and the time Δt from the warm event to the next cold event were well defined (see Fig. 7). Figure 8 demonstrates that there is a positive correlation between the size of the warm event and the time to the next cold event. Table 1 shows how the strength of this timing relationship depends on filtering—it begins to degrade when the filter is not strong enough (e.g., the 13-month running mean case in Table 1) and too much noise is left in the signal to accurately specify the extrema and Δt. The x(t) column of the last four rows of Table 1 also shows that for various low-pass x(t) series with less noise than the 13-month running mean, there is a stronger relationship between the warm episode extremum and the time to the next cold episode than the cold event extremum and the time to the next warm episode. This difference is not as strong for the commonly used Tahiti minus Darwin SOI and the NINO3.4 El Niño index (see the last two columns of Table 1).

Notice that the above timing/amplitude relationship is only true on average, so it may not hold in individual cases. Also, it is only true for the low-passed time series. ENSO time series have both a quasi-biennial and lower frequency component (see, e.g., Rasmusson et al. 1990) and the filtering emphasizes the lower frequency component.

Clarke and Li (1995) argued that the relationship between the warm event extremum and the time to the next cold event was due to eastward displacement of deep atmospheric convection and westerly wind anomalies. A larger warm ENSO episode implies that the warm pool and the westerly zonal wind anomaly is farther to the east. Consequently, the oceanic Rossby waves that the wind anomaly generates take longer to propagate to the western boundary, reflect, and return as an equatorial Kelvin wave to the region of the wind anomaly. But one expects that, when this wave transit time Δ increases, the time Δt taken to replace the westerly wind anomaly with an easterly one will also increase. Therefore, as observed, Δt should increase when the size of the warm event increases.

Crucial in the above argument is that the time Δt between warm and cold events increases when the wave transit time Δ increases. This is consistent with the analytical result (3.15) because for nondecaying solutions, ∂T/∂Δ > 0 (see appendix B); that is, the period T and, hence, Δt increases when Δ increases.

But the above analytical result assumes that the constant Δ theory is a good guide to results when Δ varies continuously as x(t) varies. When Δ is a function of x(t) as in (2.5), does the solution for x(t) have the observed timing/extremum property? Are the changes in Δ comparable to those observed when the size of the warm episode varies realistically?

When a = b = d = 0, the solution of (4.1) is red noise and the amplitude of the oscillations varies a lot. Since red noise has larger amplitudes at lower frequencies, we might expect larger events to be associated with longer times between events, consistent with the timing/extremum property in Fig. 8. Tables 2 and 3 show that, for the red noise case, larger warm (positive x) epsodes are positively correlated with Δt and larger cold episodes (negative x) are negatively correlated with δt; that is, the larger the warm episode the longer to the next cold episode and the larger the cold episode the longer to the next warm episode. However, for all red noise experiments conducted the correlations were typically much less than the observed correlations, especially for the warm episode amplitude to cold episode case [compare the x(t) column in Table 1 with the red noise columns in Tables 2 and 3]. In addition, while the observed Δt and δt for x(t) were typically about 2 years (corresponding to a periodicity of about 4 years), many red noise Δt and δt were much longer, (consistent with longer than interannual periodicity in red noise). Thus red noise does not explain the observed timing/extremum property.

When a, b, and d are nonzero and wave transit times δ and Δ are constant, the solution of (4.1) has variable amplitude but the time interval δt between cold and warm or Δt between warm and cold episodes is within a month or two of 2.5 years. The near constancy of Δt and δt at 2.5 years makes sense because when ε ≡ 0 and wave transit times Δ and δ are constant, the solution of (4.1) is sinusoidal with period 5 years. Consistent with Δt and δt being essentially constant as the amplitude varies, calculations show that the correlation between the size of episode extrema and the time till the next episode is negligible.

When a, b, and d are nonzero and δ and Δ depend on x(t), Table 2 shows correlations between the extremum of the warm episode and the time to the next cold episode are similar to those observed. Table 4 indicates that the regression relationship between the extremum of the warm episode and the time Δt to the next cold episode is comparable to that observed (0.53 ± 0.45 yr/1000 km). The best match between observed and average model regression coefficients occurs for the d = 0 (only western boundary reflection) case.

Comparison of the correlation results in Tables 2 and 3 shows that the relationship between the extremum of a cold event and the time δt to the next warm event is much weaker in the model. This is consistent with the observed correlations for x(t) in Table 1. We can understand this dichotomy between the Δt and δt correlations physically by considering the d = 0 (only western boundary reflection) case. In that case, when x(t) is negative, the wave transit time Δ is reduced and, for realistic parameters, the solution begins to decay. Mathematically, from (3.16) ∂σ/∂Δ > 0 when σ = 0. Consequently, for a cold episode (x < 0), Δ decreases and so must σ decrease, making it negative. Decay thus results. Physically, the edge of the warm pool is too close to the western boundary to allow sufficient wave delay for an oscillation to be set up. According to this model, the weak observed correlation between the size of the cold episode and the time to the next warm episode would be due to chance or red noise behavior. At the other extreme, one could imagine a model where d ≠ 0 and b = 0; that is, there is only reflection from the eastern boundary. In that case one would expect a larger (in magnitude) colder episode to lead to a longer time interval to the next warm episode. The observed dominance of the correlation between the warm episode extremum and Δt over cold event extremum and δt is thus consistent with the dominance of western boundary reflection in the ENSO mechanism. This conclusion is supported by the d = 0 model results giving the best match for the regression coefficients (see the preceding paragraph) and by physical arguments suggesting that d should be much smaller than b (see section 3b).

5. Concluding remarks

We have shown how a simple zonal warm pool displacement model can be used to understand certain aspects of ENSO physics. In particular, we examined the Clarke and Li (1995) observation that, on average, if the warm ENSO episode is larger, then it takes longer to the next cold ENSO episode. Physically, the bigger the warm ENSO episode, the farther the eastern edge of the warm pool is from the western oceanic boundary and therefore the longer it takes for the waves to feed back negatively on the growing central equatorial Pacific instability. When the lag in negative feedback increases, so, on average, does the time to the next cold ENSO episode. Comparison of model results with observations suggests that ocean wave reflection from the western ocean boundary is more important to the ENSO mechanism than ocean wave reflection from the eastern ocean boundary.

The above simple model suggests that the zonal displacement of the western Pacific warm pool and its overlying convection are fundamental to ENSO. To be realistic, it would seem that models should include this physics. A reasonable statistical test of a model time series is that it should reproduce approximately the observed warm episode extremum correlation with Δt and the corresponding regression coefficient.