## Abstract

Discharge and recharge of the warm water volume (WWV) above the 20°C isotherm in an equatorial Pacific Ocean box extending across the Pacific from 156°E to the eastern ocean boundary between latitudes 5°S and 5°N are key variables in ENSO dynamics. A formula linking WWV anomalies, zonally integrated wind stress curl anomalies along the northern and southern edges of the box, and flow into the western end of the box is derived and tested using monthly data since 1993. Consistent with previous work, a WWV balance can only be achieved if the 20°C isotherm surface is not a material surface; that is, warm water can pass through it. For example, during El Niño, part of the WWV anomaly entering the box is cooled so that it is less than 20°C and therefore passes out of the bottom of the box, the 20°C isotherm surface. The observations suggest that the anomalous volume passing through the 20°C isotherm is approximately the same as *T* ′_{W}, the anomalous WWV entering the western end of the box. Therefore the observed WWV anomaly can be regarded as being driven by the anomalous wind stress curl along the northern and southern edges of the box. The curl anomaly changes the WWV both by divergent meridional flow at the edges of the box and vortex stretching; that is, the Sverdrup balance does not hold in the upper ocean. A typical amplitude for the rate of change of WWV for the 5°S–5°N box is 9.6 Sv (Sv ≡ 10^{6} m^{3} s^{−1}). The wind stress curl anomaly and the transport anomaly into the western end of the box are highly correlated with the El Niño index Niño-3.4 [the average sea surface temperature anomaly (SSTA) over the region 5°S–5°N, 170°–120°W] and Niño-3.4 leads minus the WWV anomaly by one-quarter of a cycle. Based on the preceding results, a simple discharge/recharge coupled ENSO model is derived. Only water warmer than about 27.5°–28°C can give rise to deep atmospheric convection, so, unlike past discharge/recharge oscillator models, the west-central rather than eastern equatorial SSTAs are emphasized. The model consists of two variables: *T* ′, the SSTA averaged over the region of strong ENSO air–sea interaction in the west-central Pacific equatorial strip 5°S–5°N, 156°E–140°W and *D*′, the 20°C isotherm depth anomaly averaged over the same region. As in the observations, *T* ′ lags *D*′ by one-quarter of a cycle; that is, ∂*T* ′/∂*t* = *νD*′ for some positive constant *ν*. Physically, when *D*′ > 0, the thermocline is deeper and warmer water is entrained through the base of the mixed layer, the anomalous heat flux causing ∂*T* ′/∂*t* > 0. Also, when *D*′ > 0, the eastward current anomaly is greater than zero and warm water is advected into the region, again causing ∂*T* ′/∂*t* > 0. Opposite effects occur for *D*′ < 0. A second relationship between *T* ′ and *D*′ results because the water is warm enough that *T* ′ causes deep atmospheric convection anomalies that drive the wind stress curl anomalies that change the heat storage ∂*D*′/∂*t*. The atmosphere responds essentially instantly to the *T* ′ forcing and the curl causes a discharge of WWV during El Niño (*T* ′ > 0) and recharge during La Niña (*T* ′ < 0), so ∂*D*′/∂*t* = −*μT* ′ for some positive constant *μ*. The two relationships between *T* ′ and *D*′ result in a harmonic oscillator with period 2*π*/*νμ* ≈ 51 months.

## 1. Introduction

Meinen and McPhaden (2000) showed observationally that variations in the tropical 20°C isotherm depth, a proxy for the thermocline depth, consisted of two main modes—a tilting mode essentially in phase with eastern equatorial Pacific Ocean sea surface temperature (SST) and a discharge and recharge of warm water volume (WWV) above the 20°C isotherm whose time derivative is negatively correlated with eastern equatorial Pacific SST. These properties can be seen in Figs. 1 and 2 for the equatorial strip from 5°S to 5°N and the El Niño index Niño-3.4 (the SST anomaly averaged over the east-central equatorial Pacific region 5°S–5°N, 170°–120°W). The first EOF describes 51% of the variance and the second 35% so that these modes describe 86% of the variance, essentially all of the interannual variability of the equatorial thermocline.

The EOF structure function in Fig. 1a tilts upward in the central Pacific from about 160°E to 120°W, in approximately the same region as the westerly wind anomalies in Fig. 3 (about 155°E–140°W). Physically, the wind stress pushes surface water eastward and holds it there, the eastward wind force balancing the zonal pressure gradient. This balance was first established observationally for the interannual Pacific thermocline depth by Kessler (1990) and for the sea level by Li and Clarke (1994).

Mathematically this balance can be deduced from the zonal momentum equation

where *u* and *υ* are eastward and northward velocities in the *x* (eastward) and *y* (northward) directions respectively, *D/Dt* represents differentiation following the three-dimensional flow, *p* is the perturbation pressure due to the motion, *z* is distance upward from the undisturbed ocean surface at *z* = 0, *f* is the Coriolis parameter, *ρ* is the mean water density, and *X* is the eastward stress in the water due to the wind forcing. For interannual flow of large enough spatial scale, *Du/Dt* is negligible in (1.1) (see also a more detailed discussion of this approximation in section 2). Under this approximation, an integration of (1.1) from the depth *h* of the 20°C isotherm to the surface *z* = *η*, where *η* is the sea level, gives

where *τ ^{x}* is the eastward wind stress and

the approximation in (1.3) following from *η* ≪ *h*. In deriving (1.2) we have assumed that the turbulent stress at the depth of the 20°C isotherm is zero. At the equator *f* = 0, and (1.2) reduces to

that is, the wind stress forcing balances the depth-integrated zonal pressure gradient. This is the balance associated with EOF1 in Fig. 1. Note that off the equator (1.2) can be written

Physically (1.5) states that the northward depth-integrated flow is equal to the depth-integrated geostrophic velocity *p _{x}*/(

*ρf*) plus the northward Ekman transport, −

*τ*/(

^{x}*ρf*).

In the equatorial literature the balance (1.4) has often been referred to as “Sverdrup balance.” This is an unfortunate choice of words because, from (1.5), it implies that *V* = 0. But *V* is often referred to as “Sverdrup transport,” so we have a situation in which the Sverdrup transport is zero when the Sverdrup balance is satisfied! This is very confusing in some of the key papers in ENSO discharge/recharge theory—many authors state that Sverdrup balance of the form (1.4) or similar holds but then incorrectly state that Sverdrup transport *V* ≠ 0.

Returning to the observations, we note that the thermocline tilt/eastward wind stress balance in anomaly form does not completely explain the anomalous 20°C isotherm depth behavior. Specifically, the 20°C isotherm depth amplitude east of about 100°W falls rapidly (Fig. 1a) and the zonal wind stress anomaly structure function is only weakly negative. The discrepancy is probably due to the 20°C isotherm depth not being a good proxy for the thermocline depth because the satellite-derived sea level anomalies in phase with the zonal wind anomalies and Niño-3.4 do not fall rapidly east of 100°W (see Fig. 4). The sea level anomalies behave more in line with theory; their amplitude gently decreases east of about 120°W, consistent with the weak equatorial wind anomalies there (see Fig. 3).

While the first EOF mode is generally consistent with theory, there is some disagreement in the literature concerning the physics of the second EOF mode. In contrast with the first EOF, whose structure function is of opposite sign in the eastern and western Pacific, the second mode EOF has a structure function essentially of one sign right across the Pacific (cf. Figs. 1a and 2a). Rather than a thermocline tilt, it represents an anomalous storage of warm water. This WWV varies in time like the time integral of Niño-3.4 (see Fig. 2b).

The seminal papers of Jin (1997a, b) have discussed the discharge/recharge of WWV theoretically. However, those papers do not emphasize the importance of the wind stress curl for ENSO recharge/discharge. In fact, based on Jin’s parameters (see Jin 1997b), the zonal interannual wind stress has a large north–south scale, so large that the zonal interannual wind stress is essentially constant over the equatorial box bounded by 5°S and 5°N and the curl is very small. Quantitatively, for such a large north–south scale and realistic equatorial wind stress anomalies, curl anomalies at 5°S and 5°N generate changes in the WWV that are too small by a factor of more than 3.

In this paper we argue that the wind stress curl anomaly is crucial to the ENSO discharge/recharge. However, this does not mean that the meridional recharge/discharge is given by the traditional midlatitude formula for upper ocean Sverdrup transport—one must also take into account vortex stretching at ENSO time scales. We will also suggest that, in coupled discharge/recharge physics, the main ocean–atmosphere coupling occurs in the west-central equatorial Pacific rather than in the eastern equatorial Pacific as proposed in previous discharge/recharge oscillator models.

We begin the analysis of discharge/recharge physics by first presenting theory for the storage and discharge of WWV in an equatorial strip in section 2 and then testing it in section 3. The theory and observations suggest that the anomalous wind stress curl is crucial to the recharge and discharge of the WWV anomaly and that the meridional discharge and recharge is not in Sverdrup balance in any sense. Section 4 discusses a simplified discharge/recharge coupled model based on past results and the results in sections 2 and 3. Section 5 contains some concluding remarks.

## 2. Theory for anomalous storage and discharge of the WWV

As in the introduction let *h* be the 20°C isotherm depth and assume that it is a material surface. Define

where *z* is the vertical coordinate, **u** is the horizontal velocity, *η* is the sea level, and the approximation made in (2.1) is valid because *η* ≪ *h*. The continuity equation is

**∇**_{H} being the horizontal gradient operator and *t* the time.

Now consider a region *A* bounded by latitude lines *y* = *y _{S}* in the south,

*y*=

*y*in the north, the eastern ocean boundary, and a western boundary

_{N}*x*= 0 that is at 156°E because of data availability. Since 156°E from 5°S to 5°N is east of any western boundary current, the whole of region

*A*is governed by large-scale low frequency dynamics. Integrating (2.2) over the region

*A*using the divergence theorem gives

where *V _{S}* and

*V*refer to the northward component of

_{N}**U**at

*y*=

*y*and

_{S}*y*=

*y*, respectively, and

_{N}*U*is the eastward component of

_{W}**U**at

*x*= 0, the western edge of region

*A*. The lines

*y*=

*y*and

_{N}*y*=

*y*intersect the eastern ocean boundary at

_{S}*x*=

*L*and

_{N}*x*=

*L*, respectively. Equation (2.3) expresses the idea that ∂/∂

_{S}*t*∫

*, the time rate of increase in WWV over the region*

_{A}h dA*A*, is equal to the depth-integrated flow into

*A*through its northern, southern, and western boundaries.

Meinen and McPhaden (2001) and Meinen (2005) checked the balance (2.3) by calculating *U _{W}*,

*V*, and

_{S}*V*as a sum of geostrophic and Ekman transports, the geostrophic transport being determined from hydrographic observations. Such an analysis does not enable us to determine how the WWV anomaly is forced by the wind anomalies since the geostrophic anomalous currents are given rather than being related to the wind anomalies.

_{N}To understand how the discharge/recharge anomalies are forced in region *A*, we will need to consider both the zonal momentum equation (1.1) and the meridional momentum equation

where *Y* is the northward turbulent stress in the water due to the wind forcing. Our analysis will apply to large-scale flow at *y* = *y _{S}* and

*y*=

*y*; the scale is large enough that the Rossby number is small and the square of the baroclinic radius of deformation is small relative to the square of the scale of the flow. These large-scale approximations imply that the relative vorticity terms can be dropped from the governing vorticity equation (see appendix A for details) or, equivalently, that the

_{N}*x*and

*y*momentum equations may be written

Taking the curl of (2.5) by differentiating (2.5a) with respect to *x* and subtracting (2.5b) differentiated with respect to *y* results in

Using the mass continuity/incompressibility result

(2.6) reduces to

Integration of (2.8) with respect to *z* from the depth of the 20°C isotherm to the surface gives

where

and

is the vertical velocity at the 20°C isotherm depth.

An estimate of the error in neglecting **u** · **∇*** _{H}h* as compared with

*h*in (2.10a) is

_{t}*u*/(

*ωL*) ∼ 0.4 for the reasonable values:

_{x}*u*∼ 10

^{−1}m s

^{−1},

*L*= 4000 km, and

_{x}*ω*= 2

*π*/3 yr. But the error is actually much smaller than this as can be seen from the following argument. First note that at the 20°C isotherm depth the turbulence is negligible, so from (2.5) the currents there are in geostrophic balance:

Thus

In the upper equatorial Pacific Ocean the pressure behaves a lot like that of a 1.5-layer ocean in which *p* is proportional to *h* and, in that case,

Equation (2.13) would also be satisfied if, at the 20°C isotherm depth, the pressure due to the motion was *p*_{1} due to the first vertical mode. It follows from the above that the error in neglecting **u** · **∇**_{H}*h* as compared with *h _{t}* is not

*u*/(

*ωL*) but rather [(

_{x}*p*−

*p*

_{1})/

*p*][

*u*/(

*ωL*)], which is much smaller than 1 since (

_{x}*p*−

*p*

_{1})/

*p*is small at the 20°C isotherm depth.

where *f _{N}* and

*f*refer to the values of the Coriolis parameter at

_{S}*y*=

*y*and

_{N}*y*and

_{S}Note that *h* on the left-hand side of (2.15) is time-dependent and that (2.15) is a linear equation. From now on we will consider interannual and longer low frequency variability governed by (2.15) with *h*, curl *τ*, and *T _{W}* replaced by their low-frequency versions:

*h*′, curl

*τ*′, and

*T*′

_{W}.

Since *f* ≃ *βy* near the equator, at 5°S and 5°N we may write

Thus the second term inside the parentheses on the left-hand side of (2.15) can be written

To the extent that the trapezoidal rule of integration is a good approximation, the right-hand side of (2.18) is ∫* _{A} h*′

*dA*. In fact, the left-hand side of (2.18) and ∫

*′*

_{A}h*dA*are positively correlated [

*r*= 0.63,

*r*

_{crit}(95%) = 0.60; here and elsewhere

*r*

_{crit}is based on Ebisuzaki (1997)], but the amplitudes of the two time series differ. We may write, approximately, that

Based on the ratio of the standard deviations of the time series in (2.19), *α* = 0.53. Note that here and elsewhere in this paper we use the ratio of standard deviations as an estimate of the regression coefficient because it is more accurate than the standard least squares regression coefficient when both time series have comparable noise-to-signal ratios (McArdle 1988).

Since *α* ∼ 1, the term involving *α* in (2.20) *cannot* be neglected. But, from (2.19) this means that the second term on the left-hand side of the interannual version of (2.15) is nonnegligible. This in turn implies that *fh*′_{t} is nonnegligible in the interannual version of (2.14); that is, Sverdrup balance as defined by

does *not* hold.

## 3. Comparison of the theory with observations

We will check the theory of section 2 by testing the validity of the low-frequency version of (2.15). In addition to *h*, kindly provided by the Bureau of Meteorology Research Center, Australia (Smith 1995a, b), we will need observationally based anomalous wind stress curl and *T* ′_{W}. Given that wind data are often noisy and incomplete, the curl of the wind stress, involving a derivative of the wind stress, is even more noisy. We therefore decided to base our main results on the scatterometer-derived winds (available online at http://www.ifremer.fr/cersat/en/index.htm), the most complete and accurate curl data available.

The net interannual western boundary layer transport supplies *T* ′_{W}, the remaining term to be calculated in (2.15). Previous theoretical (e.g., Zebiak 1989; An and Kang 2001) and observational (Meinen and McPhaden 2001) work has suggested that this western boundary layer transport and, therefore, *T* ′_{W} contribute significantly. However *T* ′_{W} could not be estimated as directly as the other terms in (2.15) because the geostrophic ocean current estimates at 156°E, generously provided by Christopher Meinen (Meinen and McPhaden 2001; Meinen 2005), were only available at 3.5°S, 0°, 3.5°, and 6.5°N. We were concerned that these measurements might not be close enough together to resolve the meridional structure of the interannual geostrophic flow. We therefore calculated the interannual transport *T* ′_{W} in two ways. Both ways used along-track TOPEX/Poseidon altimeter measurements, available every 6–7 km, to obtain high-resolution interannual geostrophic surface flow. In one case we obtained the subsurface geostrophic flow structure by linearly interpolating and extrapolating the vertical structure of the geostrophic flow at 3.5°S, 0°, 3.5°, and 6.5°N based on observations. For the other estimate we noted that the observed vertical structure of the geostrophic flow at the four latitudes is similar to that of a first vertical mode, and we approximated the geostrophic subsurface flow structure by the first vertical mode structure calculated every one degree of latitude. Two estimates of *T* ′_{W} were obtained as a sum of each interannual geostrophic transport and the Ekman transport. Details are given in appendix B. Our two transport estimates for *T* ′_{W} differed negligibly.

Figure 5a shows monthly time series of the left- and right-hand sides of (2.15). Each monthly anomaly time series has been detrended and filtered with a Trenberth (1984) filter; this filter passes essentially no amplitude at frequencies higher than 2*π*/8 months and passes greater than 80% of the amplitude at frequencies lower than 2*π*/2 yr. The two time series are correlated [*r* = 0.63, *r*_{crit} (95%) = 0.60] but the amplitude of the right-hand side of (2.15) is too small. If *T* ′_{W} were zero, then the amplitudes of both sides would be nearly the same (Fig. 5b) and there would be a balance. But it is unlikely that observed *T* ′_{W} is negligible since we calculated it in two different ways (see section 2 and appendix B) and obtained nearly exactly the same nonnegligible result. Furthermore, *T* ′_{W} is well correlated with Niño-3.4 [*r* = 0.81, *r*_{crit}(95%) = 0.69], suggesting that we are calculating a meaningful signal. Thus it is likely that *T* ′_{w} is nonzero and our imbalance is real. The imbalance implies that our assumption that the 20°C isotherm depth is a material surface and can therefore only be changed by volume transport is wrong. It ignores the possibility that the WWV can be changed by anomalous heating and cooling. For example, if the WWV is anomalously cooled so that some of it is less than 20°C, then the WWV is decreased as the 20°C isotherm rises (i.e., there is a loss of WWV through the bottom of the WWV box). This is consistent with previous work by Meinen and McPhaden (2001) for a larger box (8°S–8°N, 156°E–95°W); they also found substantial transport across the 20°C isotherm. In addition, Wang and McPhaden (2000) and Holland and Mitchum (2003) both document interannual heating and cooling at the sea surface.

As noted by Meinen and McPhaden (2001), *T* ′_{W} is mostly due to western boundary flows. Although this transport is substantial, Fig. 5b shows that, if we put *T* ′_{W} = 0 in the anomaly version of (2.15), thus ignoring both *T* ′_{W} and anomalous flow through the bottom of the box, then (2.15) [and also (2.20)] approximately holds; that is, observed interannual WWV fluctuations can be regarded as being driven by wind stress curl anomalies.

Equation (2.20) then helps us understand the relationship of the time variation of the warm water volume anomaly with respect to El Niño. The wind responds rapidly to the SST anomalies, so we might expect the wind stress curl anomaly to vary in time like the SST anomaly index Niño-3.4. Figure 6 shows that this is indeed the case, as the wind stress curl anomaly term in (2.20) and Niño-3.4 are maximally correlated [*r* = 0.87, *r*_{crit} (95%) = 0.68] at zero lag. It follows from (2.20) that the WWV anomaly ∫* _{A} h dA* should be proportional to −∫

^{t}

_{0}NINO3.4(

*t*

_{*})

*dt*

_{*}. This is approximately the case; the correlation between these two time series is

*r*= 0.77 with

*r*

_{crit}(95%) = 0.69.

Why, physically, should the above correlation be negative? When there is an El Niño, the major wind anomalies in the west-central equatorial Pacific are westerly and decay away from the equator (Fig. 3). This results in a positive curl at 5°N and a negative curl at 5°S (Fig. 7). The positive curl at 5°N increases the ocean’s angular momentum in two ways [see (2.14)]: it increases its vorticity by discharging parcels northward where *f* is larger and increases its moment of inertia by decreasing *h* and thereby fattening fluid parcels. Both of these ocean adjustments lead to decreased *h*. By similar arguments, the negative curl along 5°S also decreases *h*. Thus, when there is an El Niño (Niño-3.4 is positive), the wind stress curl anomalies cause a discharge (*h _{t}* < 0) of WWV in accordance with (2.20).

## 4. A simple discharge/recharge oscillator theory

The discharge/recharge oscillator of Jin (1997a, b) emphasizes the major role of the WWV anomaly and how that leads to a self-sustained coupled ocean–atmosphere oscillation. Discharge/recharge of equatorial WWV anomaly is also central to the model to be discussed here but it differs from Jin’s in two main ways: our model emphasizes the physics associated with the wind stress curl anomalies and also that the critical ENSO air–sea interaction takes place in the west-central equatorial Pacific (≈156°E–140°W) rather than in the eastern equatorial Pacific. The west-central equatorial Pacific is where the zonal ENSO wind anomalies driving the ocean are strongest (see Figs. 3, 7) and also where the interannual ENSO heating of the atmosphere is strongest (see Fig. 8). The strongest air–sea interaction occurs here rather than in the eastern equatorial Pacific [where the sea surface temperature anomalies (SSTAs) are larger] because the eastern equatorial Pacific is usually too cold to support deep atmospheric convection—SST usually has to be at least 27.5°–28°C before tropical large-scale deep atmospheric convection can occur (Krueger and Gray 1969; Gadgil et al. 1984; Graham and Barnett 1987).

To illustrate the main physical ideas as clearly as possible, our coupled model will consist of only two central equatorial Pacific variables: *T* ′, the SSTA, and *D*′, the 20°C isotherm depth anomaly, both averaged over the region 5°S–5°N, 156°E–140°W.

### a. T′ drives D′

One component of the model is built upon the theoretical and observational results of the earlier sections of this paper. Specifically, when there is an El Niño, the SSTA *T* ′ of our box is positive, there is anomalous deep equatorial atmospheric convection (Fig. 8), and, through physics discussed by Clarke (1994), this deep atmospheric convection generates equatorial westerly surface wind anomalies (Fig. 3). These wind anomalies decrease poleward and give rise to negative wind stress curl anomalies at 5°S and positive wind stress curl anomalies at 5°N. The wind stress curl anomalies at these latitudes will discharge warm water from the box, resulting in ∂*D*′/∂*t* < 0. If this scenario is correct, then since on an interannual time scale the atmospheric response to *T* ′ is effectively instantaneous, the anomalous curl should be in phase with *T* ′. But ∂*D*′/∂*t* < 0 is driven by the anomalous curl, so we should have

for some positive constant *μ*. By similar arguments, when *T* ′ < 0, ∂*D*′/∂*t* is positive, so (4.1) is applicable for both positive and negative *T* ′. We checked (4.1) using monthly *D*′ and *T* ′ available from December 1981 to July 2005. After detrending and Trenberth (1984) filtering, the left- and right-hand sides of (4.1) were significantly correlated [*r* = 0.73, *r*_{crit} (95%) = 0.42] with the ratio of standard deviations giving *μ* = 24.6 m (yr^{−1}) (°C)^{−1} (see Fig. 9a).

The above result is built on wind stress curl anomalies causing a discharge of WWV from the box. More precisely, by analogy with (2.15) and the discussion of section 3, the WWV balance for our box is

where *x* = 0 corresponds to 156°E, *x = E* to 140°W, *T* ′_{W} the eastward transport through 156°E, *T* ′_{E} the eastward transport through 140°W, and *B* is the box bound by the surface, the 20°C isotherm, 156°E, 140°W, 5°S, and 5°N. We analyzed (4.2) in the same way as (2.15) and again found that the left- and right-hand sides did not balance but would approximately balance if *T* ′_{W} − *T* ′_{E} were omitted (see Fig. 10). Thus, as for (2.15), we may regard the observed interannual WWV fluctuations to be driven by the wind stress curl anomalies. Analogous to (2.15), (2.19), and (2.20), the left-hand side of (4.2) is proportional to ∂*D*′/∂*t*, so, with *T* ′_{W} − *T* ′_{E} omitted, we have a balance between ∂*D*′/∂*t* and the curl anomaly.

### b. D′ drives T′

Equation (4.1) describes how *T* ′ drives interannual variations in *D*′. But, we also argue below that *D*′ can drive interannual variations in *T* ′. We begin with the mixed layer heat balance equation, which for a surface mixed ocean layer of depth *H* and temperature *T* is given by [see, e.g., Wang and McPhaden (2000), their (2)]

In (4.3) *c _{p}* is the specific heat at constant pressure,

*Q*

_{0}the net surface heat flux across the air–sea interface, −

*Q*

_{pen}the heat loss by shortwave radiation that penetrates through the mixed layer, and

*Q*the vertical heat flux into the mixed layer through the base of the mixed layer. Observational analysis of the interannual version of this equation by Wang and McPhaden (2000) suggests that the anomalous heat storage term

_{W}the change in anomalous mixed layer heat due to anomalous zonal advection of the mean zonal SST gradient

and the anomalous heat flux through the base of the mixed layer *Q*′_{W} are large terms in the interannual heat balance. We therefore simplify the interannual version of (4.3) divided by *ρc _{p}H* to

Consider the first term on the right-hand side of (4.4). Jin and An (1999) argue that *u*′ is proportional to *D*′ because *u*′ is related to the meridional gradient of the thermocline depth via geostrophic balance. Since ∂*T*/∂*x* is negative and we expect *u*′ to be positively correlated with *D*′, it follows that −*u*′∂*T*/∂*x* should be proportional to a positive constant times *D*′.

With regard to the second term on the right-hand side of (4.4), first note that correlation calculations with detrended and Trenberth filtered *D*′ and ∂*T* ′/∂*t* monthly time series show that they are significantly positively correlated [*r* = 0.79, *r*_{crit}(95%) = 0.42]. This result, (4.4), and −*u*′∂*T*/∂*x* proportional to *D*′ suggest that *Q*′_{W} is also proportional to *D*′. Why should this be so?

Consider, for example, the idealized case when the mixed layer depth is constant and water is stratified beneath the mixed layer. When *D*′ = 0, normal wind-generated turbulence at the base of the mixed layer results in cooler water parcels moving into the mixed layer and warmer parcels leaving it; that is, there is a heat flux out of the mixed layer but no net exchange of mass. This mean heat flux is balanced by mean heat fluxes into the mixed layer from the surface and/or horizontal advection so that *T* ′ = 0. Suppose now that *D*′ < 0; that is, the 20°C isotherm has now been displaced upward. Since the mixed layer depth is constant, conservation of mass requires that in the nonmixing region beneath the mixed layer and above the 20°C isotherm, water must have diverged horizontally. Thus there is cooler water at the base of the mixed layer, so, even though the wind stress may be unaltered, wind-generated turbulence at the base of the mixed layer causes an anomalous heat flux down through the bottom of the mixed layer; that is, *Q*′_{W} < 0. Similar arguments suggest that, if *D*′ > 0, then *Q*′_{W} > 0. It follows that *Q*′_{W} should be positively correlated with *D*′.

Based on the above arguments, (4.4) will be written

for some positive constant *ν*. The ratio of the standard deviations of Trenberth-filtered ∂*T* ′/∂*t* and *D*′ give *ν* ≈ 0.089°C (yr)^{−1} m^{−1} (see Fig. 9b).

The relationship (4.5) is similar to that used by Jin (1997a, b) in his original discussion of a simple discharge/recharge oscillator model. Equation (4.5) differs in that *T* ′ represents the west-central equatorial Pacific temperature anomaly between 156° and 140°W whereas in Jin’s model *T* ′ refers to the SSTA in the eastern half of the basin. Note, however, that when *T* ′ refers to the SSTA in the eastern half of the basin the relationship (4.5) is no longer valid. This is because, at ENSO periodicity, (4.5) predicts that *T* ′ lags the local 20°C isotherm depth by several months, but analysis of observations by Zelle et al. (2004) shows that the lag in the eastern equatorial Pacific is a few months or less (see Fig. 11).

### c. Solution

with

For the parameter values *μ* = 24.6 m (yr)^{−1} (°C)^{−1} and *ν* = 0.089 °C (yr)^{−1} m^{−1},

a reasonable interannual frequency. Here *D*′ oscillates at the same frequency and (4.5) suggests that *T* ′ should lag *D*′ by ¼ period, that is, about 1 yr. Similarly, from (4.1), *D*′ should lag −*T* ′ by about 1 yr. These results are consistent with the lead–lag correlation calculations in Fig. 9c.

The physics of the oscillation is summarized in Fig. 12. We begin our discussion at the height of an El Niño (upper left panel). Then *T* ′ is maximally positive and the wind anomalies are maximally eastward. As shown in Figs. 1 and 2, the wind anomalies cause a twofold ocean response. One part of the response consists of an anomalous tilt of the thermocline in phase with the westerly wind anomalies as the wind stress forcing is balanced by the zonal pressure gradient. While this tilt affects the thermocline depth in the eastern and western equatorial Pacific, the net displacement in the west-central Pacific region, 156°E–140°W, is mainly due to the anomalous wind stress curl, which causes poleward transport of warm water [mathematically, when *T* ′ is a maximum the loss of equatorial warm water is greatest in (4.1)]. One quarter of a period later (Fig. 12, upper right panel), the warm water has been discharged from the equator and the thermocline is anomalously shallow (*D*′ < 0). A raised thermocline implies colder water nearer the surface and a negative heat flux through the base of the mixed layer. It also results in an anomalous westward flow that advects cold water from the eastern equatorial Pacific. These processes both cause *T* ′ to decrease [see (4.4) and (4.5)]. Eventually, one-quarter of a period later (Fig. 12, bottom left panel) *T* ′ has reached its negative extremum and the model exhibits La Niña conditions with an anomalous thermocline tilt upward in the eastern equatorial Pacific and heat content anomaly zero. Through their curl, the easterly wind anomalies cause a transport of water onto the equator so that one-quarter period later (Fig. 12, bottom right panel) the warm water volume on the equator is at its maximum and *D*′ > 0. This, then, results in a positive heat flux at the base of the mixed layer and eastward equatorial current anomalies, causing ∂*T* ′/∂*t* to be greater than zero [see (4.4) and (4.5)] and El Niño conditions to return one-quarter of a period later (Fig. 12, upper left panel).

The above idealized model errs in several respects. For example, observations (Meinen and McPhaden 2000) suggest that the same amplitude positive and negative WWV anomalies produce different amplitude SSTAs whereas the idealized model makes no such distinction. Kessler (2002) also found that the recharge/discharge oscillator only operates for three-quarters of the cycle; there is a break in the cycle from La Niña to recharge. In addition, our idealized oscillator is not phase locked to the seasonal cycle; in the observations Niño-3.4 tends to have largest variance in December and minimum variance in April. Nevertheless, the idealized model does illustrate the idea that wind stress curl anomalies play a key role in ENSO dynamics; without wind stress curl, heat storage is negligible and the recharge/discharge oscillator mechanism fails completely.

## 5. Concluding remarks

Our analysis leads us to the following main conclusions: First, near-equatorial wind stress curl anomalies play a fundamental role in ENSO dynamics since they make a major contribution to the WWV anomalies. While the wind stress curl anomalies drive the poleward transport of warm water, the midlatitude Sverdrup balance, in which poleward transport is in quasi-steady balance with the windstress curl, does not hold. This is because on ENSO time scales the vortex stretching term proportional to *h _{t}* contributes significantly in the vorticity equation and is an order-one contributor to the WWV anomalies. We also point out that it is confusing to call the balance (1.4) “Sverdrup balance” because it implies that the “Sverdrup transport”

*V*is negligible.

Second, we emphasize that unlike previously proposed recharge/discharge physics, the observations suggest that the main region of ENSO air–sea coupling is in the west-central equatorial Pacific. An idealized simple model for this region (5°S–5°N, 156°E–140°W) illustrates the essential elements of the discharge/recharge physics there.

## Acknowledgments

We are grateful for funding from the National Aeronautics and Space Administration (Fellowship NNG04GQ75H, for Giuseppe Colantuono) and the National Science Foundation (Grant ATM-0326799). Drs. Christopher Meinen (ocean current estimates at 156°E) and Neville Smith (20°C isotherm depth) generously provided data. Satellite estimates of sea level and wind stress were obtained from NASA and European Remote Sensing Web sites. The Climate Diagnostics Center, Boulder, Colorado (see online at http://www.cdc.noaa.gov/), provided the Reynolds Optimal Interpolation SST data.

## REFERENCES

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### APPENDIX A

#### Justification for the Neglect of Relative Vorticity at Low Latitude

In this equation *ζ* is the vertical component of the relative vorticity.

First note that, under the large-scale approximation that the Rossby number is small, (*ζ* + *f* ) may be replaced by *f* in (A.1). At latitude 5°, | *f*| ≃ 1.2 × 10^{−5} s^{−1} and |*ζ*| ≲ 10^{−6} s^{−1}, so this condition is satisfied.

Second, observe that the first two terms on the left-hand side of (A.1) are of the same order, so both terms are negligible if *Dζ*/*Dt* is negligible. At El Niño frequencies *Dζ*/*Dt* ≲ *ζ _{t}*, so the first two terms on the left-hand side of (A.1) are negligible if

*ζ*=

_{t}*υ*−

_{xt}*u*is negligible when compared with

_{yt}*f*(

*u*+

_{x}*υ*) at 5°N and 5°S. Because the low-frequency flows have a large zonal scale relative to the meridional scale and because

_{y}*υ*∼

*p*/

_{x}*ρf*,

*u*∼ −

*p*/

_{y}*ρf*,

*υ*/

_{xt}*u*∼ (

_{yt}*y*− scale)

^{2}/(

*x*−scale)

^{2}≪ 1. Thus at 5°S and 5°N

*υ*−

_{xt}*u*will be negligible when compared with

_{yt}*f*(

*u*+

_{x}*υ*) if

_{y}*u*is. An estimate of

_{yt}*f*(

*u*+

_{x}*υ*) can be obtained by assuming that it is dominated by the first few vertical modes. For a given vertical mode having vertical mode gravity wave speed

_{y}*c*,

It then follows from *u* ∼ −*p _{y}*/

*ρf*that

where Δ*y* is the distance of 5° latitude from the equator. For the first vertical mode *c* ≈ 2.7 m s^{−1} and the ratio is 15%, while the second vertical mode has *c* = 1.6 m s^{−1} and ratio 5%. Since the second vertical mode contribution is at least as large as the first, the ratio in (A.3) is small. Note that *c*/*f* is the baroclinic radius of deformation and Δ*y* the length scale of the large-scale flow, so the ratio on the right-hand side of (A.3) is the square of the baroclinic radius of deformation to the square of the length scale of the large-scale flow.

### APPENDIX B

#### Estimating T ′W, the Eastward Interannual Transport above the 20°C Isotherm between 5°S and 5°N at 156°E

Integration of (2.5b) from *z* = −*h* to *z* = 0 at 156°E shows that

that is, *U _{W}* is the sum of the depth-integrated eastward geostrophic flow and the eastward wind-driven Ekman transport. As noted by Meinen and McPhaden (2001), who also used (B.1), the Ekman relation fails near the equator when

*f*approaches zero. Consequently, like them, we only calculated Ekman transports poleward of 2°N and 2°S. Meinen and McPhaden note that wind-driven frictional currents equatorward of 2° are shallow and weak, having small transports compared with geostrophic transports (McPhaden 1981; Picaut et al. 1989). We calculated the eastward Ekman transport using the scatterometer wind dataset mentioned in section 3. Monthly transport anomalies were determined by calculating the average transport for each calendar month and then subtracting this 12-point annual time series from the original monthly time series. These current anomalies were then filtered with the Trenberth (1984) filter (see description in section 2) to obtain monthly interannual current anomalies.

As noted in section 3, monthly geostrophic ocean current estimates at 156°E were generously made available at 3.5°S, 0°, 3.5°, and 6.5°N by Christopher Meinen (Meinen 2005). These estimates were derived using the meridian 156°E of the equatorial Pacific TAO/TRITON Array (see online at http://www.pmel.noaa.gov/tao/index.shtml). We calculated monthly interannual geostrophic currents from the monthly time series by removing the 12-point annual time series and Trenberth filtering in a similar way to the Ekman transport.

An EOF analysis of these filtered geostrophic current anomalies at each of the four moorings down to the average 20°C isotherm depth *h* shows that they are well described by the first EOF at each of the four moorings (see Fig. B1). Thus we may write that the interannual geostrophic velocity

with the *G _{i}*(

*z*) being the EOF functions in Fig. 13. These functions were normalized, without loss of generality, such that

*G*(0) is unity. We suppose that an EOF analysis at other latitudes between 5°S and 5°N would, like those at 6.5°N, 3.5°N, 0°, and 3.5°S, also be strongly dominated by the first EOF mode. Then

_{i}with *G* unity at *z* = 0. This implies that *ϕ*(*y*, *t*) is the surface geostrophic velocity. By analogy with (2.1)

where

Note that, strictly speaking, *d*(*y*) should involve an integral from the actual depth *h* of the 20°C isotherm rather than the time-averaged depth *h*(*y*), but the error in replacing *h* by *h* is negligible.

From (B.4) and the definition of *T* ′_{W} we have

where *U* ′_{Ek} is the interannual Ekman transport (*τ ^{Y}*)′/(

*ρf*). As noted earlier,

*U*′

_{Ek}is taken to be zero between 2°N and 2°S.

If *d*(*y*) and *ϕ*(*y*, *t*) were adequately resolved by the measurement array at the four latitudes, 6.5°N, 3.5°N, 0°, and 3.5°S, then we could approximate the first term on the right-hand side of (B.6) by a finite sum. But calculations show that the *ϕ _{i}*(

*t*) are insignificantly correlated with each other (Table B1) and

*d*(

*y*) is considerably larger south of the equator (see Fig. B2) where we only have one measurement.

_{i}We overcame the *ϕ* resolution problem by calculating the eastward interannual surface flow *ϕ* from high-resolution along-track satellite altimeter data for the 2 nearest tracks to 156°E. In each case we assumed that the zonal scale was much larger than the meridional so that dynamically both tracks were perpendicular to the equator and high-resolution zonal interannual geostrophic surface currents could be estimated. Our final estimate was an average of the estimates for each track.

With regard to the *d*(*y*) resolution problem, we noticed that the structures *G _{i}*(

*z*) in Fig. 13 are similar to those of the first vertical mode, suggesting that high-resolution

*G*(

*y*,

*z*) in (B.3) and hence

*d*(

*y*) in (B.5) might be obtained by calculating the first vertical mode as a function of latitudinally varying buoyancy frequency and water depth. Figure B2 shows an estimate of

*d*(

*y*) obtained from (B.5) and such an estimate for

*G*together with

*d*(

*y*) based on the EOF

_{i}*G*(

_{i}*z*) from (B.2). We calculated

*T*′

_{W}using

*d*(

*y*) in (B.6) from a first vertical mode calculation and

*d*(

*y*) based on the interpolation and extrapolation of the four EOF

*d*(

*y*) values from the estimated currents. The two

_{i}*T*′

_{W}time series differed negligibly.

## Footnotes

*Corresponding author address:* Allan J. Clarke, Department of Oceanography 4320, The Florida State University, Tallahassee, FL 32306-4320. Email: clarke@ocean.fsu.edu