Abstract

Atmosphere–ocean coupled processes responsible for generating and maintaining the equatorial warm pool were investigated using models of different complexities. The primary focus was to answer the following question: why is the observed warm pool concentrated around the maritime continent? In this first of a two-part series, the solutions of a simple conceptual model that represents the tropical Pacific and Indian Oceans interacting via the Walker circulation are examined. When the interbasin coupling is sufficiently strong, surface wind convergence over the Maritime Continent associated with easterly trades over the Pacific acts to generate the equatorial westerly over the Indian Ocean, leading to a warm pool spontaneously emerging between the two ocean basins. The conceptual model shows that tropical climate has two equilibria, depending upon the ocean basin widths—a single warm pool regime corresponding to the current climate and a split warm pool regime that accompanies warm pools created in the western parts of each ocean basin. The latter is found to be unstable and hence exhibits large-amplitude vacillations between the ocean basins being further amplified by the Bjerknes feedback. The above two regimes of the equatorial warm pool are identified in the model incorporating the interactive Atlantic Ocean as well, wherein the mean state and variability in the three ocean basins qualitatively agree with the observations.

1. Introduction

Despite the fact that the insolation at the top of the atmosphere is independent of longitude, tropical climate as represented by the sea surface temperature (SST) has considerable zonal nonuniformity. In particular, the existence of the warm pools, which drive the general circulation of the atmosphere, is peculiar; they are characterized by regions of relatively uniform SST, higher than 28°C, spreading around the Maritime Continent and the western tropical Atlantic Ocean (Fig. 1, top). Two fundamental properties of the warm pools that have been discussed in the literature are the spatial homogeneity and the longitudinal placement; the latter contrasts with the presence of cold tongues in the eastern basin (e.g., Clement et al. 2005).

Fig. 1.

Annual mean climatology of (top) SST, (middle) subsurface temperature, and (bottom) zonal stress along the equator. The contours of 26°C (28°C) are indicated in the top panel.

Fig. 1.

Annual mean climatology of (top) SST, (middle) subsurface temperature, and (bottom) zonal stress along the equator. The contours of 26°C (28°C) are indicated in the top panel.

The spatial homogeneity of the warm pool, as frequently represented by the negatively skewed probability distribution of the tropical SST, suggests that processes that prohibit an increase in the SST are at work. The primary candidate is the cirrus cloud, which reflects the solar radiation and hence acts as a thermostat for the ocean surface (Ramanathan and Collins 1991; Ramanathan et al. 1995). Subsequently, the importance of other processes was suggested, such as negative feedbacks resulting from, for example, surface evaporation (Hartmann and Michelsen 1993) and tropospheric relative humidity (Pierrehumbert 1995). All of these processes are atmospheric, so that a bulk argument regarding the atmospheric net heat transport regulating the warm pool SST is also possible (Wallace 1992); this may, in turn, be constrained by the oceanic heat transport, mostly resulting from the Ekman drift (Held 2001).

Apart from its homogeneity, the warm pool is regarded as the surface manifestation of warm water piled up in the equatorial upper ocean (Fig. 1, middle). This necessarily accompanies a zonal tilt in the thermocline, which is deeper (shallower) in the west (east) both in the Pacific and the Atlantic Oceans. The tilt is also associated with the existence of cold tongues in the eastern basin. Over the past decades, it has been elucidated that this climatological state is responsible for El Niño–Southern Oscillation (ENSO). In short, a perturbed wind stress gives rise to changes in ocean upwelling and thermocline tilt, both of which favorably affect the cold tongue SST. The resultant change in the zonal SST gradient works to intensify the initial wind stress anomaly by modulating the atmospheric zonal differential heating. This is the crux of the Bjerknes (1969) feedback on which the ENSO amplification relies. The thermocline change is dominated by the slow ocean dynamical adjustment, which provides the time scale of ENSO (Neelin et al. 1998, and references therein).

Not only ENSO but also the climatological east–west contrast in the SST and thermocline depth can now be understood with the same mechanism, sometimes called the “climatological version” of the Bjerknes feedback (Neelin and Dijkstra 1995; Jin 1996, hereafter J96). Dijkstra and Neelin (1995), using an intermediate coupled model for the tropical Pacific, have shown that the warm pool–cold tongue pattern varies considerably in the model with different sets of parameters, such as the efficiency of the upwelling and the thermocline on SST and the magnitude of the wind stress response to the zonal SST gradient, all of which control the climatological Bjerknes feedback. J96 has demonstrated that a coupled system that is similar to but simpler than that of Dijkstra and Neelin (1995), which is also used in this study, can explain both the climatological state and ENSO in terms of the bifurcation of the equilibrium solution from stable to unstable branches.

The role of the ocean dynamical adjustment has been discussed even in terms of regulating the warm pool SST to an upper bound (Sun and Liu 1996; Clement et al. 1996; Seager and Murtugudde 1997). Li et al. (2000) and Clement et al. (2005) quantified the relative role of the feedbacks resulting from cloud, evaporation, and ocean dynamics using an intermediate coupled model. Except for some differences in the detail, both studies conclude that all of those processes are important for regulating the warm pool SST.

Observations reveal that the Pacific warm pool smoothly extends into the eastern Indian Ocean (cf. Fig. 1). This feature has not been focused in detail in the above studies, with a few exceptions (e.g., Schneider et al. 1996). The diabatic heating over the Maritime Continent driving the easterly winds over the Pacific would necessarily force the equatorial westerlies over the Indian Ocean, which in turn helps maintain the warm pool by preventing the equatorial upwelling in the eastern Indian Ocean. The observed warm pool is thus better viewed as the surface warm water piled from both sides of the Maritime Continent resulting from the cooperative wind stresses over the western Pacific and the eastern Indian Ocean. The coupling between these two ocean basins has recently been extensively investigated in conjunction with the interannual variability associated with ENSO (Klein et al. 1999; Lau and Nath 2000, 2003; Watanabe and Jin 2002, 2003). Several studies using coupled general circulation models (CGCM) further showed that the Indian Ocean is active in modulating the ENSO amplitude and periodicity (Yu et al. 2002; Wu and Kirtman 2004; Kug et al. 2006). These studies may advocate the necessity of dealing with the coupled atmosphere–ocean systems in the Pacific and Indian Oceans together for understanding the climatological mean state as well.

In this study, we attempt to demonstrate that the warm pool around the maritime continent is the product of the tropical atmosphere–ocean system in the Pacific and Indian Oceans as a whole, rather than the dynamical adjustment in the Pacific alone. Furthermore, the extent to which the widely spread, single warm pool as observed is a unique solution in the coupled system is investigated by exploring the equilibrium or climatological states in a hierarchy of models, in which the tropical Pacific and Indian Oceans are as idealized as possible without losing generality. In Part I herein, a conceptual model for tropical oceans coupled with the Walker circulation is used to draw an outline of the hypothesis proposed throughout this study. A more comprehensive coupled model will be used in the companion paper (Watanabe 2008, hereafter Part II).

This paper is organized as follows: In the next section, a conceptual model for the tropical atmosphere–ocean system is described. The model that has been proposed to explain the mean state and its variability primarily in the tropical Pacific is revised in this study. In section 3, equilibrium states of the simple tropical climate model are examined. Their stability and dependence on several control parameters are also presented. Section 5 presents a summary, discussion, and remarks leading to Part II.

2. A simple tropical model

We use a simple tropical climate model proposed by J96. This model has two degrees of freedom for the ocean mixed layer temperature, or simply the SST, in the eastern basin Te, and the departure of the thermocline depth from a reference depth in the western basin hw. The J96 model is regarded as a prototype to the so-called recharge oscillator model (Jin 1997), which is a well-known theoretical model for ENSO but has an advantage that the solutions are obtained for the mean state but not for the anomaly from the mean state.

The eastern basin SST is controlled by

 
formula

where

 
formula

The first term on the right-hand side of (1) represents the net heating resulting from radiative and surface heat fluxes while the second term represents the dynamical cooling resulting from the equatorial upwelling. The net heating is simply expressed by the restoration of the SST toward a zonally uniform, radiative–convective equilibrium temperature Tr, with the time scale ɛ−1T. The dynamical cooling originates from the vertical advection at the base of the mixed layer, so that it depends on the subsurface temperature Tse, prescribed mixed layer depth Hm, and the upwelling velocity w. Note that the function θ(w) results from the upstream differencing, indicating that the upwelling cools the mixed layer, but not vice versa.

The upwelling velocity is solely due to Ekman pumping, proportional to the zonal wind stress τ along the equator,

 
formula

where αd can be determined by the oceanic momentum dissipation rate and β at the equator (Zebiak and Cane 1987). The zonal stress can be divided as follows:

 
formula

where τ0 is the stress resulting from the Hadley circulation and is therefore defined to be negative, while the second term is associated with the Walker circulation, which is driven by the SST contrast between the eastern (Te) and western (Tw) basins. The latter is assumed to be in the radiative–convective equilibrium, namely, Tw = Tr. The relevance of this assumption will be discussed in section 4b. This part, when coupled with the thermocline change, is a key component of the climatological Bjerknes feedback. The parameter μ is the feedback coefficient that measures the efficiency of the zonal differential heating in driving the Walker circulation.

The subsurface temperature is strongly influenced by the thermocline depth in the eastern basin and may be parameterized as

 
formula

where Tr0 is the temperature beneath the thermocline, z0 is the depth at which w takes its characteristic value, and h* is a measure of the thermocline sharpness. The thermocline depth in the eastern basin is given by H + he, where he is the departure from the reference depth H and is determined by the integral of the Sverdrup relationship,

 
formula

where bLτ is proportional to the zonally integrated wind stress, and hence L is the ocean basin width and b is a parameter for the efficiency of wind stress in creating the thermocline tilt. The thermocline depth in the western basin hw adjusts slowly to the zonally integrated Sverdrup mass transport that is proportional to the zonally integrated wind stress curl resulting from the basin-wide dynamical adjustment at a rate r,

 
formula

where the second term on the right-hand side represents the zonally integrated Sverdrup mass transport, to which a factor r/2 has been multiplied so that equilibrium hw and he are − bLτ/2 and bLτ/2, respectively. The derivation and rationales of (6) have been given in more detail in J96.

Equations (1)(6) form a closed set in which the prognostic variables are only Te and hw. Following J96, the constants are set as follows: Tr = 303 K, Tr0 = 291 K, ɛT = (75 days)−1, r = (300 days)−1, Hm = 50 m, H = 100 m, z0 = 75 m, h* = 50 m, μ = 0.1 N m−2 K−1, and μαd = 7.5 m K−1 month−1. The primary control parameters in this simple model are the coupled feedback coefficient, the ocean basin width, and the zonally uniform part of the wind stress, which are all changeable for different ocean basins. In J96, they are combined as μbL and τ0/μ, which are 12.5 m K−1 and −1 K for the tropical Pacific, 6.25 m K−1 and −1 K for the tropical Atlantic, and 6.25 m K−1 and +1 K for the tropical Indian Ocean. The stationary solutions can be calculated for each ocean and collectively plotted in Fig. 2a, which reproduces Fig. 1 of J96. In the equilibrium state, thermodynamic balance is achieved between the surface net heating and the dynamical cooling resulting from upwelling [right-hand side of (1)]; the latter feature is different from basin to basin (Fig. 2a).

Fig. 2.

(a) Stationary solutions to the simple coupled model. Rates of the net heating (thin straight line with reversed sign) and dynamical cooling (thick curves) are plotted against Te under the dynamical equilibrium. The stationary state is indicated by the intersection points. Note that the three curves mimicking the Pacific, Atlantic, and Indian Oceans, respectively, are separately calculated. (b) Same as (a), but for the solutions with the interbasin coupling between the Pacific and the Indian Oceans (solid) compared with the solution without coupling (dotted).

Fig. 2.

(a) Stationary solutions to the simple coupled model. Rates of the net heating (thin straight line with reversed sign) and dynamical cooling (thick curves) are plotted against Te under the dynamical equilibrium. The stationary state is indicated by the intersection points. Note that the three curves mimicking the Pacific, Atlantic, and Indian Oceans, respectively, are separately calculated. (b) Same as (a), but for the solutions with the interbasin coupling between the Pacific and the Indian Oceans (solid) compared with the solution without coupling (dotted).

The Bjerknes feedback is proportional to the zonal integral length of τ in (6) and to the coupled feedback in (3), so that the upwelling reduces the SST more efficiently as μbL becomes larger. The equilibrium state of Te is around 24°C in the Pacific (intersection in Fig. 2a), which indicates a large zonal contrast in the SST and the thermocline depth. When the basin length is reduced to half (or equivalently, the coupled feedback weakens), which corresponds to the Atlantic Ocean, the cold tongue becomes warmer by about 3°C, as seen in the observations. In addition, the equatorial upwelling is no longer effective in creating the cold tongue if the background wind stress τ0 is reversed; this mimics the situation in the tropical Indian Ocean.

On one hand, the J96 model captures the basic characteristics of the oceanic climatological mean states in a single set of equations. Furthermore, linearized equations for (1) and (6) exhibit that the Pacific mean state is unstable, leading to an ENSO-like oscillation, while the other two oceans are not. Premising that the coupled feedback and the ocean dynamics commonly work to maintain the mean states and to generate anomalies from them, this model provides a dynamical basis to ENSO as well as the cold tongue. On the other hand, the model may have two concerns when arguing the dynamics of the warm pool. One is that the stationary states are independent of each other between the ocean basins; this is not consistent with the descriptive view of the warm pool, as mentioned in the introduction. Another weakness is that τ0 induced by the Hadley circulation must be negative at any situation. Nevertheless, it has to be positive for the Indian Ocean; otherwise, the cold tongue emerges there.

The Walker circulation is primarily forced by convective heating over the equatorial land region; this will accompany the convergence in the lower atmosphere (e.g., Stone and Chervin 1984). This convergence can couple two ocean basins via the atmospheric dynamics, leading to the surface easterly (westerly) to the east (west) of the active convections. We therefore extend the J96 model in order to include the coupling of two ocean basins as follows. First, two ocean basins, referred to as the first (placed to the west) and second (placed to the east) basins, are defined. Most of Eqs. (1)(6) are identical between basins 1 and 2, except for the wind stress, which is now expressed as

 
formula
 
formula

where the subscripts 1 and 2 indicate the variables in basins 1 and 2, respectively, and C represents a convergence feedback that couples the two ocean basins as

 
formula

We have implicitly assumed that the presence of land gives rise to this convergence feedback, which automatically generates a positive τ1 if the Bjerknes feedback in basin 2 is strong enough. The convergence feedback is controlled by the parameter δ, which is set to δ/μ = 1.25. This value determines the relative influence between the interbasin temperature contrast and the temperature difference within the basin to the wind stress, which is changed in section 3c to explore the role of the interbasin coupling in the model solutions. Basins 1 and 2 mimic the tropical Indian Ocean and the Pacific, so we set μbL1 = 6.17 and μbL2 = 12.33 m K−1, where Li indicates the ith basin width, and τ0/μ is fixed at −1 K for both oceans. Two sets of Eqs. (1), (2), and (4)(6), in addition to (7)(9), have four degrees of freedom and are referred to as the extended J96 model.

Figure 2b shows the equilibrium states as in Fig. 2a, but for the extended J96 model with (solid) and without (dashed) the convergence feedback. In the absence of the convergence feedback, or the interbasin coupling, equilibrium Te in the Pacific, namely, Te2, is identical to that in Fig. 2a, while Te in the Indian Ocean (basin 1) reveals a weak cold tongue, which is the same as that in the Atlantic Ocean of the J96 model. The interbasin coupling not only makes the Pacific cold tongue stronger, but by reversing the wind stress in the Indian Ocean, changes the Indian Ocean SST compared to the radiative–convective equilibrium temperature. The extended J96 model is thus shown to be relevant in obtaining the warm pool between the Pacific and Indian Oceans as well as the strong cold tongue in the eastern Pacific with a reasonable definition of τ0.

3. Two regimes in the simple tropical climate model

a. Stationary solutions

As argued in J96 and also shown in Fig. 2a, as the ocean basin length increases the dynamical cooling works more effectively in creating the cold tongue in the eastern basin. In the extended J96 model, convergence occurs over the land between the two basins when the cold tongue is only slightly weak [hence, the zonal stress is also weak through (7)] in basin 1, resulting in a spontaneous emergence of the single warm pool between the two basins (Fig. 2b). It is therefore clear that the ratio between L1 and L2 is a critical parameter for the equilibrium states. To identify possible solutions of the extended J96 model, the parameter that represents the relative basin width is defined as

 
formula

The two ocean basins have equal length with α = 0 while the “Pacific” is twice as large as the “Indian Ocean” with α = 1, which is the case shown in Fig. 2b. In nature, α may be roughly estimated to be in the range of 1 ≤ α ≤ 1.5, although its precise value is hardly obtained because of the complicated land geometry. With the whole basin width LtL1 + L2 fixed at μbLt = 18.5 m K−1, stationary solutions can be presented as a function of α (Fig. 3). The model consists of a set of nonlinear equations, so that the stationary solution is first calculated at α = 0 when the two oceans are identical, that is, C = 0 in this particular solution; therefore, the equilibrium Te can be obtained as shown in Fig. 2. The stationary solution at a slightly larger α is then computed from the initial guess given by the stationary state at α = 0 using an iterative method. This procedure is repeated until α = 1.4 to find a branch originating from the stationary state for the equal basin length. Another branch is also identified by first calculating the solutions at α = 1.4 and then repeating with a reduced α. Because we attempt to mimic the coupled Indo-Pacific climate, positive α cases are examined here. However, we can use negative α as well, possibly representing the coupled Pacific–Atlantic Oceans, which will be explored in section 3d.

Fig. 3.

Stationary solutions for Te in the Pacific (circles) and Indian (squares) Oceans as a function of α with prescribed coefficients of the interbasin coupling. Search of the two branches are started from the solutions at α = 0 and α = 1.4, respectively (indicated by arrows).

Fig. 3.

Stationary solutions for Te in the Pacific (circles) and Indian (squares) Oceans as a function of α with prescribed coefficients of the interbasin coupling. Search of the two branches are started from the solutions at α = 0 and α = 1.4, respectively (indicated by arrows).

The equilibrium Te for the two basins is shown in Fig. 3, in which two branches of the stationary states are identified. One is originating from the solutions with α = 0 at which the mean states reveal a weak cold tongue that is identical between the two oceans. As α increases, the cold tongue is weakened but continues to be present in basin 1. This implies that the warm pool is split into two at the western part of each ocean basin; therefore, we call this branch the split warm pool regime. This regime does not have stationary solutions beyond α ≈ 0.7, where another branch appears, that is, the strong cold tongue in the Pacific, accompanied by the radiative–convective equilibrium in the Indian Ocean, which are qualitatively the same as the state shown in Fig. 2b and are here denoted as the single warm pool regime. In the range of 0.35 < α < 0.7, we also find multiple regimes, one of which is actually selected depending on the initial state.

The stability of the steady mean states shown in Fig. 3 is first examined by simply integrating the extended J96 model in time with an initial state slightly deviating from the equilibrium state. The trajectory for Te in the two ocean basins is plotted against α in Fig. 4a, which indicates that the whole split regime and a part of the single warm pool regime are unstable. This gives rise to the supercritical Hopf bifurcation at around α = 0.9. The trajectory in such unstable regimes is sufficiently far from the stationary state (denoted by triangles in Fig. 4a); with α = 0, for instance, the time series of Te shows that a vacillation between the strong cold tongue (Te < 22°C) and the radiative–convective equilibrium (Te = 30°C) is regularly taking place in the two oceans (Fig. 4b). In one particular ocean basin, the eastern basin SST and the western basin thermocline depth oscillate with an out-of-phase relationship, which has been found in the ENSO-like oscillatory solution of the recharge model (cf. Fig. 3b of Jin 1997), except for a much larger amplitude in the extended J96 model (Fig. 4c). This oscillation occurs during the period for which another ocean basin is in mostly the radiative–convective equilibrium state (Fig. 4d). Once the eastern basin SST approaches the radiative–convective equilibrium temperature, it stays for a longer time and another ocean basin starts to warm up. This looks like a conventional coupled oscillation between two springs, but some nonlinearity must work because the warm states are prolonged more than the cold states (Fig. 4b).

Fig. 4.

(a) Trajectory plots for Te (dots) imposed on the stationary solutions (bold triangles) as a function of α. The black (gray) marks indicate the split (single) warm pool regime. (b) Time series of Te in the Pacific (solid) and the Indian Ocean (dashed) with α = 0. (c) Trajectory plots for Te and hw in the Indian Ocean. (d) Same as (c), but for Te in the two ocean basins.

Fig. 4.

(a) Trajectory plots for Te (dots) imposed on the stationary solutions (bold triangles) as a function of α. The black (gray) marks indicate the split (single) warm pool regime. (b) Time series of Te in the Pacific (solid) and the Indian Ocean (dashed) with α = 0. (c) Trajectory plots for Te and hw in the Indian Ocean. (d) Same as (c), but for Te in the two ocean basins.

The reason why the split warm pool regimes are not stably equilibrated can be explained as follows. When basin 2 swings toward the El Niño–like state, the equatorial easterly is weakened and the resultant surface divergence anomaly over the land region, which may represent the suppressed convections there, necessarily accompanies the intensified easterly in basin 1 that eventually leads to the La Niña–like state. This coupled instability is prohibited as Te in basin 2 approaches the upper limit. As long as the Bjerknes feedbacks are similar in magnitude in the two basins, the La Niña–like state in one basin can significantly maintain the El Niño–like state in another basin. While the two basins are interchangeable in this simple model, the influence of one basin on another may not be symmetric in more complicated systems that explicitly represent the convectively coupled equatorial waves in the atmosphere. This asymmetry will be highlighted in Part II.

b. Linear stability

A complete picture of the stability of the solutions in Fig. 3 can be obtained with a linearized version of the extended J96 model. Let x be the state vector

 
formula

which consists of perturbations (denoted by prime) for the prognostic variables in two ocean basins; the linearized equations are expressed in the matrix form

 
formula

where

 
formula

The terms in (13) are as follows:

 
formula
 
formula
 
formula
 
formula

where the overbar denotes the stationary state, i is the index of ocean basins, and (x) is the Heaviside function of x. It should be noted that in the absence of the interbasin coupling, that is, δ = 0, 𝗟 is reduced to a block diagonal matrix, indicating that the two ocean basins are independent of each other.

Eigenmodes of 𝗟 are numerically obtained separately for each α. The growth rate and frequency for the most unstable modes are then plotted in a similar manner to the stationary solutions (Fig. 5). Consistent with Fig. 4a, all of the steady states in the split regime and a part of the single regime are found to be linearly unstable. The split regime is generally more unstable than the single regime because the convergence feedback can destabilize the system, as qualitatively described in the previous subsection. In the single warm pool regime, the convergence is driven solely by the wind stress in basin 2 (Pacific) because of Te1 = Tr. It is not quite clear why the most unstable mode emerges with α = 0.4, but the reason for this may be speculated as follows: The positive feedback induced by the interbasin coupling is the strongest with α = 0, while the conventional Bjerknes feedback works more efficiently with a larger α. The steady mean state may therefore be the most unstable with an intermediate value of α. In most of the single warm pool regime, the leading eigenmode is oscillatory, regardless of its growth rate, with a period between 1.9 and 2.9 yr. The oscillation at the neutral mean state with α = 0.9 has a periodicity of 2 yr, which is somewhat shorter than that in J96, but still appears to represent the prototype of ENSO. As in the J96 model, the Bjerknes feedback is reduced for a too-strong cold tongue, so that the steady states with extremely large α are stable and do not reveal any oscillatory eigenmode.

Fig. 5.

Linear stability of the simple coupled model as a function of α. The open (closed) marks indicate growth rate (frequency) for the most unstable eigenmode, obtained when the model is linearized with respect to stationary states in the split (triangle and square) and single (circle) warm pool regimes, respectively.

Fig. 5.

Linear stability of the simple coupled model as a function of α. The open (closed) marks indicate growth rate (frequency) for the most unstable eigenmode, obtained when the model is linearized with respect to stationary states in the split (triangle and square) and single (circle) warm pool regimes, respectively.

c. Sensitivity to the convergence feedback

As demonstrated previously, the convergence feedback acts to maintain the stable single warm pool for α ≥ 0.9, with the prescribed coefficient of the interbasin coupling, δ/μ = 1.25. This coefficient is, however, tunable and the equilibrium state is switched to the split regime with δ = 0 (Fig. 2b). We therefore explored how the mean state depends on this convergence feedback, which now varies from 0 to 2. The equilibrium Te with α = 1 is plotted as a function of δ/μ in Fig. 6, which again reveals the following two distinct regimes: an unstable split regime with δ/μ < 1.2 and a stable single warm pool regime with a larger δ/μ. Compared to the regime dependence on α (Fig. 3), the regime transition in Fig. 6 is smoother and no hysteresis occurs, indicating that the convergence feedback is a crucial piece in stabilizing the single warm pool regime in the simple model while the stationary solutions are more sensitive to the relative basin width than to the magnitude of the interbasin coupling.

Fig. 6.

As in Fig. 3, but for varying interbasin coupling coefficient δ/μ, with α prescribed to 1.0. The open (closed) marks indicate the stationary solutions are linearly unstable (stable).

Fig. 6.

As in Fig. 3, but for varying interbasin coupling coefficient δ/μ, with α prescribed to 1.0. The open (closed) marks indicate the stationary solutions are linearly unstable (stable).

d. Coupling with the “Atlantic Ocean”

Thus far we have used the extended J96 model with positive α, namely, we have set basin 2 equal to or greater than basin 1, which mimics the coupled Pacific and Indian Oceans. The same model but with negative α may represent the coupled Pacific and Atlantic Oceans, in which basin 1 (basin 2) will be assigned to the tropical Pacific (Atlantic). Assuming that the Pacific is twice as large as the Atlantic, the observational estimate will be α ≈ − 0.5 [cf. Eq. (10)].

The regime diagram with negative α is roughly a mirror to Fig. 3 (not shown); the single (split) warm pool regime occurs with a smaller (larger) α. The observed Atlantic climatology reveals a weak cold tongue and a small warm pool (Fig. 1), so that the split regime that exists around α = −0.4 may be of relevance. However, the split regime in the coupled Pacific–Atlantic vacillates as in the counterpart in the coupled Pacific–Indian Ocean. This is an unrealistic feature that is never observed, suggesting that the extended J96 model does not capture the observed mean state in the tropical Atlantic resulting from the lack of other important processes.

In nature, the Pacific, Indian, and Atlantic Oceans are all coupled and likely to interact with each other. Because the Pacific warm pool is much larger than that of the Atlantic, the coupling of three ocean basins may better represent the mean state in the equatorial Atlantic. We therefore coupled the third basin (basin 3), which mimics the Atlantic Ocean, to the extended J96 model and then reexamined the equilibrium states. The governing equations for basin 3 are the same as those for the other basins, in which the interbasin coupling term consists of the zonal stress difference with the other two basins. For simplicity, the width of basin 3 is prescribed as Lt/2, and the relative width between basins 1 and 2 (the Indian and the Pacific Oceans) is changed with α, as in section 3a. This experimental setup will be used in a GCM study of Part II. Note that the three basins have identical widths at α = 0.

The stationary solutions of this coupled three-ocean system are calculated with the standard magnitude of the interbasin coupling of δ/μ = 1.25 (Fig. 7a). At α = 0, the following three sets of equilibria are identified: the cold tongue is generated in all of the basins (Te = 26.6°C), one basin is in radiative–convective equilibrium while the other two basins have the cold tongue (Te = 25.1°C), and two basins are in radiative–convective equilibrium with one basin having the cold tongue (Te = 23.0°C). The first equilibrium solutions may be the “pure” split regime, where all of the ocean basins actually vacillate with the temporal phase leading/lagging by π/3 to each other; however, this state disappears at α > 0. The second solutions also correspond to the unstable split regime in which the vacillation occurs between two basins and the rest of the ocean weakly oscillates around the radiative–convective equilibrium. In these regimes, the Pacific cold tongue becomes stronger as α increases, but they switch to the third solutions corresponding to the single warm-pool regime at around α = 0.2 (Fig. 7a).

Fig. 7.

Regime diagram in the coupled three-ocean system with different convergence feedback parameters: (a) δ/μ = 1.25, (b) δ/μ = 1, and (c) δ/μ = 1 for the Indo-Pacific and δ/μ = 0.5 for the other basin coupling. Stationary solutions for Te in the Pacific (circles), Atlantic (triangles), and Indian (squares) Oceans are plotted as a function of α. Stable (unstable) solutions are indicated with sold (open) marks.

Fig. 7.

Regime diagram in the coupled three-ocean system with different convergence feedback parameters: (a) δ/μ = 1.25, (b) δ/μ = 1, and (c) δ/μ = 1 for the Indo-Pacific and δ/μ = 0.5 for the other basin coupling. Stationary solutions for Te in the Pacific (circles), Atlantic (triangles), and Indian (squares) Oceans are plotted as a function of α. Stable (unstable) solutions are indicated with sold (open) marks.

Compared to the two-ocean system (Fig. 3), the three-ocean system has more equilibria, among which the single regime is the dominant state. It is also found, by comparing Figs. 3 and 7, that the interactive Atlantic works to stabilize the system in order to maintain the strong Pacific cold tongue. In this regime, however, the Atlantic Ocean is yet in radiative–convective equilibrium. When the interbasin coupling that also acts to stabilize the system (Fig. 6) is slightly weakened to δ/μ = 1, the stable regime is confined to α ≥ 0.45, in which the Atlantic Te decreases from Tr by 2°–3°C. This indicates the presence of the weak cold tongue (Fig. 7b; note that the second solutions are not plotted to prevent the diagram from being too cluttered).

Assuming that the wind convergence induced by the SST contrast is less effective over a wide continent, such as Africa and South America, we then reduced δ/μ to half that between the Indo-Atlantic and the Pacific–Atlantic basins. The regime diagram obtained with the above condition is displayed in Fig. 7c. The two regimes in the Indo-Pacific Ocean are quite similar to those found in Fig. 3, and they accompany the weak cold tongue in the Atlantic. One difference from the two-ocean system is that both the regimes are linearly unstable for 0 ≤ α ≤ 1.4, which may be rather reasonable because the stable single regime cannot generate ENSO autonomously (cf. J96). In fact, the time evolution of Te in the three oceans at α = 1.2 in the model shown in Fig. 7c reveals the interannual variability prevailing within a 3–5-yr period in both the Pacific and the Atlantic (Fig. 8). They are reminiscent of the observed ENSO and the “Atlantic Niño” (e.g., Zebiak 1993) accompanying the Pacific warm state that intermittently bursts to almost 30°C when the Indian Ocean shows a weak cold state. This will be an indication of a ghost of an unstable split regime, which may provide an alternative mechanism of the El Niño bursting (Timmermann et al. 2003).

Fig. 8.

Time evolution of Te in the Pacific (thick solid), Atlantic (thin solid), and Indian Ocean (dashed) at α = 1.2 in the experiment shown in Fig. 7c.

Fig. 8.

Time evolution of Te in the Pacific (thick solid), Atlantic (thin solid), and Indian Ocean (dashed) at α = 1.2 in the experiment shown in Fig. 7c.

4. Summary and discussion

a. Summary

Motivated by the fact that the Pacific warm pool spreads to the eastern Indian Ocean, and also by recent studies that show the active role of the Indian Ocean SST variability in the Pacific ENSO cycle, this study aims at understanding the possible role of the interaction between the two oceans in establishing the climatological mean warm pool. In this first of a two-part series, we extended Jin’s (1996) simple coupled model for one ocean basin in order to incorporate the coupling of two oceans via the Walker circulation; we then explored the equilibrium states and their stability by varying the parameters of the interbasin coupling and the relative ocean width.

When the interbasin coupling is sufficiently strong, surface convergence over the Maritime Continent associated with the easterly winds over the Pacific acts to generate the equatorial westerly over the Indian Ocean, which gives rise to a spontaneous emergence of the warm pool between the two ocean basins. Furthermore, the simple system suggests that the tropical climate has two equilibria, depending on the ocean basin widths. One is the single warm pool regime corresponding to the current climate and the other is the split warm pool regime that accompanies warm pools created in the western side of each ocean basin. While the split regime can occur when two oceans are relatively close in basin width, such a regime is found to be unstable because of the interbasin coupling, and hence exhibits a large-amplitude vacillation between ocean basins, each of which is amplified by the Bjerknes feedback. The above two regimes of the equatorial warm pool are also identified in the coupled three-ocean system that includes the Atlantic Ocean.

b. Discussion and remarks leading to Part II

The presence of multiple equilibria of the tropical atmosphere–ocean system has been reported in previous studies (e.g., J96; Sun and Liu 1996; Timmermann et al. 2003) in the absence of the interbasin coupling. In the J96 model, for example, if the coupled feedback parameter μbL is eventually reduced from the value for the Atlantic (6.25 m K−1), we obtain two equilibria of the weak cold tongue state and the radiative–convective equilibrium in the entire ocean, with the latter being linearly unstable (cf. Fig. 2a). The two regimes of the warm pool obtained from the extended J96 model are conceptually different from such results dealing with one ocean basin because the convergence feedback is crucial in producing the single warm pool regime. When we change μ in our extended J96 model, as has been done in simple model studies for ENSO, a stable branch of the radiative–convective equilibrium state is obtained. Such solutions are, however, trivial and independent of the interbasin coupling.

One of the controversial simplifications of the model may be the western basin SST, which is assumed to be always in radiative–convective equilibrium (cf. section 2). Because of this, the Indian Ocean SST in the single warm pool regime is uniform throughout the basin in the simple model, which may be too crude to represent the observed Indian Ocean climatology. In reality, the superrotating surface wind induces the upwelling in the western tropical Indian Ocean, slightly south of the equator, which results in a coherence between the subsurface temperature and the SST (Xie et al. 2002). A simple model also shows that the cold tongue in the western basin possibly appears with a certain value of the coupling coefficient (Dijkstra and Neelin 1995). We therefore examined the equilibrium solutions in the model in which the western basin SST Tw is allowed to vary due to the upwelling, as in (1). The results did not alter the conclusions obtained here, namely, that the steady mean states were quite similar to those shown in Fig. 3, except for the equilibrium Tw in the Indian Ocean that was cooler by 1–2 K than Tr only in the single warm pool regime. As will be shown in Part II, the results in more complicated coupled model are indeed consistent with the solutions of the extended J96 model, which further justifies that the assumption can be crudely accepted.

Interbasin coupling appears to have two different roles in the regime behavior of the conceptual model. The coupled vacillation between the two basins found in the split regime is crucially dependent upon the interbasin coupling, which actually favors the single warm pool regime (Fig. 2b). In fact, the stronger the convergence feedback, the wider the range of α the stable single regime occupies (Figs. 6 and 7a,b). The stable regime, however, may not be realistic because it cannot generate an ENSO-like interannual variability without an external forcing. While the question of whether ENSO is a self-sustained oscillation or a stochastically excited variability may still be controversial, the unstable single regime with moderate strength of the interbasin coupling seems to match the observations in terms of the mean state and the interannual variability (Figs. 7c and 8). As presented in section 3d, the interactive Atlantic Ocean exhibits a richer behavior of the equilibrium solutions than the coupled Indo-Pacific system shown in sections 3a–c. While the overall regimes are similarly identified both in the two- and three-ocean systems, coupling with the interactive Atlantic does affect the regime transition and stability, as will be shown with a CGCM in Part II.

The presence of regimes in simple tropical climate models, such as the one used herein, crucially depends on how to represent the nonlinear ocean dynamic feedback associated with the thermocline tilt and the upwelling. There is some arbitrariness in this regard, so that the results presented here must be validated using a model having much larger degrees of freedom. We have done this work in Part II.

Acknowledgments

The author is grateful to Drs. F.-F. Jin and A. Timmerman for stimulating discussions and constructive comments. Thanks are also due to Drs. J. P. McCreary, S.-P. Xie, N. Schneider, B. Qiu, J.-S. Kug, and anonymous reviewers who provided valuable comments. This work was partly supported by a Grant-in-Aid for Scientific Research from MEXT, Japan.

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Footnotes

Corresponding author address: M. Watanabe, Center for Climate System Research, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8568, Japan. Email: hiro@ccrs.u-tokyo.ac.jp