In this section, we describe the components of the hybrid coupled model which has been developed in this project with the objective of exploring the role of the southward wind shift in the synchronization of ENSO events to the seasonal cycle.
Ocean model
The ocean model utilised here is a shallow-water model (SWM), whose name refers to the fact that the horizontal scale of the planetary scale waves (100–1000 km) is much larger than the vertical scale (ocean depth \(\sim\)4 km), which allows the Navier-Stokes equations to be simplified considerably. It is a linear reduced-gravity model resolved on a 1° × 1° spatial grid for the low- to mid-latitude global ocean between 57°S–57°N and 0°–360°E. The density structure of the 1\(\frac{1}{2}\)-layer baroclinic system consists of a well mixed active upper layer of uniform density overlaying a deep motionless lower layer of larger uniform density. These ocean density layers are separated by an interface (the pycnocline) that provides a good approximation of the thermocline. This is a crucial consequence as it allows us to quantify the upper-ocean heat content (e.g. Rebert et al. 1985), i.e., the warm-water volume (Meinen and McPhaden 2000), and provide an estimate of equatorial SSTA (e.g., Kleeman 1993; Zelle et al. 2004).
The ocean dynamics are described by the linear reduced-gravity form of the shallow-water equations detailed below [Eqs. (2)–(4)]:
$$\begin{aligned} u_{t}-fv+g'\eta _{x}&=\frac{\tau ^x}{\rho H}+F_m \end{aligned}$$
(2)
$$\begin{aligned} v_{t}+fu+g'\eta _{y}&=\frac{\tau ^y}{\rho H}+F_m \end{aligned}$$
(3)
$$\begin{aligned} g'\eta _t+c_1^2(u_x+v_y)&=0 \end{aligned}$$
(4)
where u and v are the eastward and northward components of velocity respectively (m s\(^{-1}\)), t is time (s), H represents the mean pycnocline depth, H = 300 m (Tomczak and Godfrey 1994, p. 37), f (s\(^{-1}\)) is the Coriolis parameter, \(\rho\) is the ocean water density, \(\rho\) = 1000 kg m\(^{-3}\), and \(F_m\) the bottom friction per unit mass. The reduced gravity, \(g'\), reflects the density difference between the upper and lower layers. We use the typical value of \(g'\) = 0.026 m s\(^{-2}\) (Tomczak and Godfrey 1994, p. 37). The corresponding first baroclinic mode gravity wave speed, \(c_1 = \sqrt{g' H})\), is 2.8 m s\(^{-1}\). The long Rossby wave speed \(C_R\) (m s\(^{-1}\)) is given by the equation, \(C_R\) = \(\beta (c_1^2/f^2)\), where \(\beta\) (m\(^{-1}\) s\(^{-1}\)) is the derivative of f northward.
The model time step is 2 h and Fischer’s (1965) numerical scheme is utilized for model time stepping. Motion in the upper layer is driven by the applied wind stresses (per unit density), \(\tau\) (m\(^2\) s\(^{-2}\)), which are anomalies from long-term monthly means (i.e., seasonal cycle removed). The associated response of the ocean is displayed by the vertical displacement of the thermocline, \(\eta\) (m), and the horizontal velocity components (u and v) of the flow velocity. This model formulation permits Ekman pumping and both Rossby and Kelvin wave propagation along the thermocline to be generated with appropriate large-scale wind stress forcing. It also includes realistic continental boundaries that were calculated as the location where the bathymetric dataset of Smith and Sandwell (1997) has a depth of less than the model mean thermocline depth of 300 m.
Regarding the calculation eastern-central Pacific SSTA, we utilise a simplified version Kleeman’s (1993) SST equation by applying the thermocline anomaly term only. Kleeman (1993) shows that this single term is primarily responsible for hindcast skill in ENSO predictions. Thus, while being the simplest scheme, it contains the essential physics required to produce realistic SSTA. Hence, the equatorial SSTA depends only on the thermocline depth anomaly. Changes in the SSTA on the equator are modeled by the equation
$$\begin{aligned} T_{t} = \alpha (x) \eta (x)-\epsilon T \end{aligned}$$
(5)
where T is the SSTA at time t, \(\epsilon\) is the Newtonian cooling coefficient, \(\epsilon\) = 2.72 × 10\(^{-7}\) s\(^{-1}\), x is the longitude and \(\alpha\) is a longitude-dependent parameter that relates the modeled oceanic thermocline depth displacement \(\eta\) along the equator to the SSTA, being α = 3.4 × 10\(^{-8}\) °C m\(^{-1}\)s\(^{-1}\) in the eastern Pacific and reducing linearly west of 140°W to a minimum of \(\alpha\)/5 at the western equatorial boundary at 120°E. Such a difference reflects the fact that the equatorial thermocline depth anomalies display a tighter connection with SSTA in the east than the west (Zelle et al. 2004). For the rest of latitudes, a fixed meridional structure that decays away from the equator with an e-folding radius of 10° is assumed. Taking into account the non-linear relationship between central Pacific zonal wind stress anomalies and Niño-3 index as reported by Frauen and Dommenget (2010), the parameter \(\alpha\) is reduced by 20 % for negative SSTA in Niño-3 region. In addition, a threshold of 37.5 m is set on the maximum absolute depth of equatorial thermocline anomalies in order to prevent runaway coupled instability.
It is also worthwhile to mention that the use of this simplified SST equation implies that each of these HCMs can generate only EP El Niño and La Niña events, i.e. only one EOF of SSTA. Therefore, the results of these HCM simulations will not distinguish between EP-CP event differences. It has been documented in several studies that this ocean model can produce observed variations of ocean heat content and sea surface heights reasonably well (e.g., McGregor et al. 2012a, b). Furthermore, a validation of this ocean model was carried out by simply forcing the model with ERA-interim monthly wind stress anomalies over 1979–2013. The modeled Niño-3 and Niño-3.4 indexes were then compared with those observed during the same period revealing correlation coefficients of 0.83 and 0.82, respectively (statistically significant above the 99 % level).
Statistical atmospheric model
The statistical atmosphere has been constructed by the two leading EOFs of wind stresses over the tropical Pacific. It has been shown above that the linear combination of both EOFs can reproduce quite well the southward shift of the maximum westerly wind anomalies and its related seasonal weakening of equatorial westerly wind anomalies, both of which have been proposed to contribute to the transition between El Niño and La Niña (e.g., Harrison and Vecchi 1999; Vecchi and Harrison 2003, 2006; Lengaigne et al. 2006; McGregor et al. 2012a, 2013).
The statistical atmospheric model is coupled to the ocean SWM to produce three hybrid coupled models (HCM): HCM1 consists of EOF1 only (i.e., no meridional wind movement); HCM1+2 and HCM1+2\(_{\mathrm{S}}\) include both EOF1 and EOF2 (i.e., they both produce meridional wind movement). In all cases, the EOF1 coupling is achieved by modelling the EOF1 surface wind stress response by:
$$\begin{aligned} (\tau _1^x,\tau _1^y) = PC1(t) \times (EOF1^x,EOF1^y) \end{aligned}$$
(6)
where PC1 is approximated by the modeled Niño-3 index. The close relationship between these two variables was noted earlier.
The method used to calculate PC2 in HCM1+2 is a least squares second-order polynomial fit from PC1 for each calendar month (month),
$$\begin{aligned} (\tau _2^x,\tau _2^y) = PC2(PC1,month) \times (EOF2^x,EOF2^y) \end{aligned}$$
(7)
where we use the two closest months to our month of interest (e.g., data taken for February, includes January and March also) in order to obtain a smooth transition of PC2 values from one month to another (Fig. 4). The second-degree polynomial function is of the form,
$$\begin{aligned} PC2 = a \cdot PC1^2 + b \cdot PC1 \end{aligned}$$
(8)
where a and b depend on calendar month. The small independent term is set to zero in order to remove any seasonal cycle in EOF2. A full list of quadratic polynomial coefficients as well as their correlation coefficients and RMSE for each calendar month are given in Table 1.
The method used to calculate PC2 in HCM1+2\(_{\mathrm{S}}\), on the other hand, is based on a climate mode that emerges through the atmospheric non-linear interaction between ENSO and the annual cycle known as C-mode (Stuecker et al. 2013, 2015). Here PC2 wind stresses are calculated by,
$$\begin{aligned} (\tau _2^x,\tau _2^y) = PC2_{S} \times (EOF2^x,EOF2^y) \end{aligned}$$
(9)
where \(PC2_S = PC1(t) \times \cos (\omega _a\,month - \varphi )\) refers to PC2 simple, which comes from the lowest-order term of the atmospheric nonlinearity. Here \(\omega _a\) denotes the angular frequency of the annual cycle, \(\omega _a=2\pi /12\) rad month\(^{-1}\) and \(\varphi\) represents a one-month phase shift, \(\varphi =2\pi /12\) rad. How well observed data fit this HCM for each calendar month is indicated by RMSE and correlation coefficients in Table 1.
It is clear that the relationship between PC1 and PC2 values depends strongly on calendar month (Fig. 4). The relationship between the pair is quasi-linear during JJA, with increasing values of PC1 being related to decreasing values of PC2. The relationship during DJF, on the other hand, displays a clear non-linearity with PC2 values increasing for increasing positive values of PC1, while the PC2 amplitude also appears to increase for decreasing negative values of PC1. Thus, the seasonal difference between the relationship between PC1 and PC2 is most pronounced for strong El Niño events (high values of PC1). Such behaviour is represented reasonably well by the HCM1+2 configuration (Fig. 4); for instance, for strong El Niño events (\(2<\) PC1\(< 3\)), PC2 prior to the event peak (JJA) has values around minus unity, while around the event peak (DJF) PC2 is between two and three, which is consistent with the sign change shown in Fig. 3a. Interestingly, however, such a strong seasonal change is not observed in moderate El Niño events (PC1 ~1) and La Niña events (PC1\(< 0\)), which is consistent with the ENSO phase and type asymmetry reported by Lengaigne et al. (2006). This ENSO phase and type non-linearity is not represented, however, in HCM1+2\(_{\mathrm{S}}\) where the relationship between PC2 and PC1 is linear regardless the calendar month (Fig. 4). Thus, the HCM1+2 simulations only have a weak southward wind shift during La Niña events, while the HCM1+2\(_{\mathrm{S}}\) simulations have a strong southward wind shift and the magnitude of the easterlies are also stronger.
Reconstructing PC2 with the polynomial fit of HCM1+2 and comparing with PC2 from the observations reveals a correlation coefficient of 0.61, while doing the same analysis for the HCM1+2\(_{\mathrm{S}}\) reconstructed PC2, reveals a correlation coefficient of 0.42. Thus, here we consider HCM1+2 as the more realistic experimental set up and HCM1+2\(_{\mathrm{S}}\) as the idealized southward wind shift, with RMSE 0.66 and 0.70 in JJA; and 0.83 and 1.20 in DJF, respectively (see Table 1 for the rest of calendar months). However, due to lack of data for strong negative SSTA over the eastern equatorial Pacific for our analysis period, we take both methods into consideration in order to examine the sensitivity of the HCM results.
Westerly wind burst model
Westerly wind activity has been shown to play an important role in the onset of El Niño events (Latif et al. 1988; Kerr 1999; Lengaigne et al. 2004; McPhaden 2004). These wind events, known as westerly wind bursts (WWB), force downwelling Kelvin waves, which propagate to the eastern equatorial Pacific and ultimately act to warm SST there, potentially initiating the event (e.g., Giese and Harrison 1990, 1991). Equatorial westerly wind activity has been associated with tropical cyclones (Keen 1982), cold surges from midlatitudes (Chu 1988), the convectively active phase of the Madden–Julian oscillation (Chen et al. 1996; Zhang 1996), or a combination of all three (Yu and Rienecker 1998).
Although different definitions have been proposed to diagnose WWB from observations (e.g., Harrison and Vecchi 1997; Yu et al. 2003; Eisenman et al. 2005), there is a broad agreement that it can be represented roughly by a Gaussian shape in both space and time,
$$\begin{aligned} u_{\mathrm {wwb}}(x,y,t) = A exp\left( -\frac{(t-T_0)^2}{T^2} -\frac{(x-x_0)^2}{L_x^2} -\frac{(y-y_0)^2}{L_y^2}\right) \end{aligned}$$
(10)
where \(x_0\) (160°) and \(y_0\) (0°) are the central longitude and latitude of the wind event, \(T_0\) (10 days) is the time of peak wind, A is the peak wind speed, T (10 days) represents the event duration, and \(L_x\) (20°) and \(L_y\) (9°) are the spatial scales. The values of these parameters are set here to obtain realistic values of wind stresses over the western Pacific (Niño-4 region). In regards to their frequency, Eisenman et al. (2005) found and average of 3.1 westerly wind events (WWEs) per year during 1990–2004, Gebbie et al. (2007) identified an average of 3.6 WWEs per year during 1979–2002 and Verbickas (1998) found 3.8 WWEs per year during 1979–1997.
In Sect. 5 we incorporate WWB into the HCM by utilising the WWB equation above, and having the probability of a WWB beginning on any given day set a fixed parameter which depends on the simulation set up. This means that we have WWBs that are purely stochastic, with the different parameter choice simply modulating the rate of WWB occurrence and their magnitude. Although it has been increasingly recognized that WWB are partially modulated by the SST field and partially dependent upon stochastic processes in the atmosphere (e.g., Kessler and Kleeman 2000; Eisenman et al. 2005; Gebbie et al. 2007), here WWB are represented by purely stochastic way due to the simplicity of our SSTA formulation (Gebbie and Tziperman 2009). Nevertheless, this paper is not about the response of El Niño events to different flavours of WWB. Rather, the intent of this paper is to focus on the role of the southward wind shift on the termination of ENSO events.