The Dynamics of ENSO Phase Locking in a Spatiotemporal Oscillator Model

YaoKun Li aCollege of Global Change and Earth System Science, Faculty of Geographical Science, Beijing Normal University, Beijing, China

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Abstract

The ENSO phase locking to the annual cycle is investigated by applying a spatiotemporal oscillator (STO) model, in which the annual cycle of the climatological thermocline depth and its associated parameter are introduced. It is easy to derive its analytic solution, which demonstrates a harmonic oscillation of a combined variable. The ENSO phase locking can be theoretically proven by discussing the distribution of the calendar months of the peak time of the sea surface temperature anomaly (SSTA) time series. The calendar months of the peak time can be divided into two parts. The first part can evenly distribute in any a month of a year and hence has no phase locking feature whereas the second part, directly associated with the annual cycle, adds an increment onto the first part to make it move toward the phase of the annual cycle to realize the phase locking feature. This is the physical mechanism of the ENSO phase locking. With observed seasonal variation of the climatological thermocline depth, the Niño-3.4 index time series approach to extreme values in November was calculated with higher probability, reproducing the observed phase locking phenomenon quite well. The maximum probability of the calendar month that the ENSO peak time occurs is directly determined by the phase of the annual cycle and the stronger the annual cycle is, the larger the maximum probability is.

Significance Statement

El Niño–Southern Oscillation (ENSO) events tend to be strongest in the boreal wintertime. This phenomenon is called ENSO phase locking. This study investigates the dynamics of ENSO phase locking to the annual cycle by introducing the annual cycle to a spatiotemporal oscillator (STO) model that can deal with the complex spatial and temporal variations in SSTAs. The analytic solution can be obtained and then the phase locking feature can be theoretically proven and numerically testified. Therefore, the dynamics and the mechanism of ENSO phase locking can be comprehensively understood. It may be beneficial for the community to have a better understanding of this complex phenomenon.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: YaoKun Li, liyaokun@bnu.edu.cn

Abstract

The ENSO phase locking to the annual cycle is investigated by applying a spatiotemporal oscillator (STO) model, in which the annual cycle of the climatological thermocline depth and its associated parameter are introduced. It is easy to derive its analytic solution, which demonstrates a harmonic oscillation of a combined variable. The ENSO phase locking can be theoretically proven by discussing the distribution of the calendar months of the peak time of the sea surface temperature anomaly (SSTA) time series. The calendar months of the peak time can be divided into two parts. The first part can evenly distribute in any a month of a year and hence has no phase locking feature whereas the second part, directly associated with the annual cycle, adds an increment onto the first part to make it move toward the phase of the annual cycle to realize the phase locking feature. This is the physical mechanism of the ENSO phase locking. With observed seasonal variation of the climatological thermocline depth, the Niño-3.4 index time series approach to extreme values in November was calculated with higher probability, reproducing the observed phase locking phenomenon quite well. The maximum probability of the calendar month that the ENSO peak time occurs is directly determined by the phase of the annual cycle and the stronger the annual cycle is, the larger the maximum probability is.

Significance Statement

El Niño–Southern Oscillation (ENSO) events tend to be strongest in the boreal wintertime. This phenomenon is called ENSO phase locking. This study investigates the dynamics of ENSO phase locking to the annual cycle by introducing the annual cycle to a spatiotemporal oscillator (STO) model that can deal with the complex spatial and temporal variations in SSTAs. The analytic solution can be obtained and then the phase locking feature can be theoretically proven and numerically testified. Therefore, the dynamics and the mechanism of ENSO phase locking can be comprehensively understood. It may be beneficial for the community to have a better understanding of this complex phenomenon.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: YaoKun Li, liyaokun@bnu.edu.cn

1. Introduction

The El Niño–Southern Oscillation (ENSO) cycle, a fluctuation between unusually warm (El Niño) and cold (La Niña) conditions in the tropical Pacific, is the most prominent year-to-year climate variation on Earth (McPhaden et al. 2006). Since ENSO events are closely tied to seasonal cycle, the peak sea surface temperature anomalies (SSTAs) tend to occur in December or January (Rasmusson and Carpenter 1982). This feature of preferred peak timing, known as the phase-locking phenomenon, is still an active subject in ENSO research (Chen and Jin 2021).

The phase locking of ENSO has been extensively investigated. Nonlinear resonance with Earth’s annual cycle is thought of as one of the possible mechanisms (Jin et al. 1994; Tziperman et al. 1994). For example, Neelin et al. (2000) suggested that the nonlinear interaction of the ENSO cycle with the annual cycle tends to the phase-locking behavior. They further suggested that the frequency is nonlinearly adjusted so that the phase matches the favorable season if the inherent frequency of an ENSO event is relatively easy to match an integer year; and the ENSO warm phase will not be perfectly locked to the preferred season but rather scattered around it if there is a compromise between the inherent cycle and the annual cycle. The annual cycle in the climate background is also deemed as another possible mechanism. Jin et al. (1996) argued that annual cycle can lock the period to exactly two years by linear mechanisms if the period of ENSO mode is close to two years. Thompson and Battisti (2000) suggested that the annual cycle in the basic state of the ocean is sufficient to produce strong phase locking of ENSO to the annual cycle by applying singular vector analysis and Floquet analysis on a linearized variant of the Zebiak–Cane (ZC) atmosphere–ocean model. They further suggested that it is not necessary to invoke either nonlinearity or weather noise. Stein et al. (2010) found that the seasonal synchronization of events is set by ENSO’s growth rate parameter by constructing a linear, stochastic oscillator model that is based on the simplest form of the recharge oscillator (RO) model. By multiplying the annual cycle to a modified harmonic oscillator model, An and Jin (2011) obtained the linear solutions of the first-order approximation for the frequency and amplitude modulation of ENSO using the perturbation method. Stein et al. (2014) further suggested that the seasonal varying background state of the equatorial Pacific on ENSO’s coupled stability, rather than the periodic forcing by the annual cycle, is responsible for the phase locking of ENSO to the annual cycle. Chen and Jin (2020) further comprehensively revisited the phase locking of ENSO by utilizing the simple RO paradigm.

Factors that have a significant annual cycle may insert their influence on the phase locking phenomenon. For example, Tziperman et al. (1997) highlighted the seasonal cycle in the equatorial Pacific, including the wind divergence field that is determined by the seasonal motion of the intertropical convergence zone, the seasonality of the background SST and ocean upwelling velocity. An and Wang (2001) suggested that the seasonal variation of the western Pacific surface wind anomalies and the mean thermocline depth are critical. Vecchi and Harrison (2006) suggested that the influence of the annual cycle of the insolation is fundamental. Dommenget and Yu (2016) suggested seasonal changes of the surface shortwave radiation associated with cloud-cover feedback is contributive. Wengel et al. (2018) stressed the importance of the equatorial cold SST bias. Besides, orbital parameters can also modulate the ENSO seasonal phase locking, which is shifted periodically following the precessional forcing: ENSO tends to peak in boreal winter when perihelion is near the vernal equinox, but to peak in boreal summer when perihelion lies in between the autumnal equinox and winter solstice (Lu and Liu 2019).

Previous investigations have been greatly prompted by the desire to understand the mechanisms of phase locking characteristics. Note that they basically apply either the coupled air–sea interaction models or simple RO paradigm by introducing annual varying parameters. Recently, Li (2024) constructed a spatiotemporal oscillator (STO) model that is based on the thermodynamics and thermocline depth dynamics. The analytic solution of the model suggests that the spatiotemporal variations in SSTA can be decomposed into a series of eigen spatial modes that oscillate with their natural frequencies. Each of the eigenmodes can be considered as a harmonic RO model. The STO model provides a systematic view of the complex spatial and temporal variations in ENSO events. Therefore, it is natural to ask whether the STO model can reproduce the phase locking phenomenon or not. In this paper, the author attempts to investigate the question by introducing seasonal cycle on parameters in the STO model. The results suggest that the simple STO model does have the capacity to capture the main phase locking characteristics.

The paper is arranged as follows. Section 2 describes the STO model with annual varying parameters and derives its analytic solution. Then the phase locking feature is theoretically proven. Section 3 introduces the observed annual cycle of the climatological thermocline depth. Section 4 analyzed the ENSO phase locking feature in the STO model. The analysis is based on the main eigenmodes and their combinations with different weights to feature different spatial patterns of ENSO: the eastern Pacific (EP) El Niño and central Pacific (CP) El Niño. Section 5 discusses the influence of the amplitude and the phase of the annual cycle on the ENSO phase locking and further clarifies the physical mechanism of the ENSO phase locking. Section 6 summarizes the investigation and provides some discussion.

2. Model description

a. The parametric spatiotemporal oscillator model

The conceptual model this paper utilizes is the simple STO model (Li 2024). Its governing equations are listed below:
Tt+u¯Tx+Khh=Q,
ht+u¯hx+Hux=0,
u=bTx,
where T indicates SSTA; u¯ is the zonal mean ocean current in the mixed layer and u is the zonal ocean current anomaly; Q is the net radiation anomaly and is generally replaced by a Newtonian cooling term −αT where α > 0 is the damping coefficient; h is the thermocline depth anomaly (TDA); H is the climatological mean depth of the mixed layer; and b is a coupling coefficient between the zonal ocean current anomaly and SSTA gradients. Equation (3) demonstrates that there exists a westerly (an easterly) wind anomaly west (east) of a warm SSTA and hence, a positive (negative) zonal ocean current anomaly west (east) of a warm SSTA in the mixed layer. This is because the ocean current in the mixed layer generally follows the direction of the wind stress. Kh is a coefficient that parameterizes vertical upwelling and mixed layer processes (Battisti 1988). It should be noted that here in this investigation, the positive parameter Kh actually plays a role in determining the oscillation frequency, rather than in determining the stability of the system. It can also be seen as a parameterization of the recharge/discharge terms in the RO model (Jin et al. 2020).

The thermodynamics Eq. (1) is commonly applied in theoretical analysis (Hirst 1986, 1988; Battisti and Hirst 1989; Kang and An 1998; Kang et al. 2001). The thermocline dynamics Eq. (2) is also commonly applied in coupled air–sea interaction models (Cane and Zebiak 1985; Zebiak and Cane 1987; Hirst 1988; Neelin et al. 1992; Jin and Neelin 1993; Neelin and Jin 1993) that also introduce atmospheric and oceanic motion equations for enclosing the system. Here the approximation relation Eq. (3) simplifies the system by avoiding introducing the motion equations for the zonal wind and zonal ocean current. Therefore, the STO model has a solid physical basis and the explicit explanation can be seen in its original form (Li 2024).

To reproduce the phase locking characteristics, annual varying parameters should be considered. Let us first consider the parameter Kh, which is found to be the most sensitive to the background states such as the mean depth of the mixed layer H (Hirst 1986). On the other hand, the mean depth of the mixed layer H has significant seasonal variation, that is also thought to be critical for the phase locking (An and Wang 2001). Therefore, it is reasonable to assume that the mean depth of the mixed layer and the parameter Kh follow the same annual cycle, that is,
H=H0A(t)H0[1μsin(ωatϕ)]
and
Kh=Kh0A(t)Kh0[1μsin(ωatϕ)],
where H0 and Kh0 are the climatological mean mixed layer depth and Kh, respectively; A(t) = 1 − μ sin(ωatϕ) represents its dimensionless seasonal variation with an annual frequency ωa=(2π/12)month1 and a phase ϕ. The term μ is a parameter to control the amplitude of the seasonal variation. The bigger it is, the larger amplitude the seasonal variation has. Here a reasonable range for μ is (0, 1) to make sure both H > 0 and Kh > 0.
With the annual varying parameter, the governing equations can be rewritten as
Tt+u¯Tx+Kh0A(t)h=0,
ht+u¯hx+H0bA(t)2Tx2=0.
Note that the net heat flux forcing is neglected here for simplicity. However, the net heat flux forcing actually plays a role. For example, the seasonal phase lock of the surface shortwave radiation associated with cloud-cover feedbacks can also contribute to the phase locking of ENSO (Dommenget and Yu 2016). According to Li (2024), the net heat flux can embody its influence on the natural oscillations as Eqs. (6) and (7) suggested by introducing forced oscillations. This investigation mainly emphasizes the phase locking of ENSO natural oscillations and it also can be extended to a more general form in the future to identify the influence of the net heat flux forcing.
Eliminating h from the system, it is easy to derive a wave equation:
(t+u¯x)2T1A(t+u¯x)A(t+u¯x)Tc02A22Tx2=0,
where c0=Kh0H0b is the climatological mean propagation speed of the wave. Introducing variables
x1=xu¯t;t1=t
to transform Eq. (8) to a simpler form, we get
2Tt121AdAdt1Tt1c02A22Tx12=0.
Equation (10) is a standard wave equation with varying coefficients. To derive its analytic solution, the initial values are specified as
T(x1)|t1=0F(x1),
h(x1)|t1=0G(x1),
where F and G are known functions. The well-posed free boundary conditions are set to
Tx1|x1=0=0,Tx1|x1=L=0,
where L is the basin width of the equatorial Pacific.

b. The cosine series expansion solution

Although Eq. (10) contains varying coefficients, it can still be variable separated. Therefore, it will be easy to derive its solution by applying cosine series expansion, e.g.,
T=n=0Tn(t1)cos(λnx1),
where λn=nπ/L, n = 0, 1, 2, …. It is obvious that Eq. (14) naturally satisfies the boundary condition Eq. (13). Substituting Eq. (14) into Eq. (10), it is easy to derive
d2Tndt121AdAdt1dTndt1+ωn2A2Tn=0,
where ωn = c0λn is the natural frequency. If n = 0, Eq. (15) can be reduced to
d2T0dt121AdAdt1dT0dt1=0,
and it is easy to find its solution,
T0=(1+C00)t1+μωacos(ωat1ϕ)+C0,
where C00 and C0 are two integral constants. To make sure T0 is limited when t1 tends to be infinity, the integral constant C00 = −1. According to the initial values, C0=f0(μ/ωa)cosϕ. Therefore, Eq. (17) finally becomes
T0=μωacos(ωat1ϕ)+f0μωacosϕ,
where f0 is the zeroth-order expansion coefficients for initial SSTA value. Equation (18) shows that the 0th-order mode of cosine expansion for SSTA has an obvious annual cycle that is the same as the ones with the parameters. It represents the annual cycle of SSTA.
If n > 0, Eq. (15) is a second-order ordinary differential equation with varying coefficients. Its analytic solution can still be derived although it looks a little complex. Introducing the variable
τ=t1+μωacos(ωat1ϕ),
it is easy to transform Eq. (15) to
(dτdt1)2d2Tndτ2+[(d2τdt12)1AdAdt1dτdt1]dTndτ+ωn2A2Tn=0.
Note that by applying the relations
dτdt1=1μsin(ωat1ϕ)A,
d2τdt12=μωasin(ωat1ϕ)dAdt1,
Eq. (20) can be reduced to a second-order ordinary differential equation with constant coefficients
d2Tndτ2+ωn2Tn=0,
and its analytic solution can be immediately written out as
Tn=Cnsin[ωnt1+μωnωacos(ωat1ϕ)+θn],
where Cn=an2+bn2 and sinθn=an/Cn are determined by the initial values, an=fnq+(Kh0gn/ωn)p, bn=fnp(Kh0gn/ωn)q, p=sin[μ(ωn/ωa)cosϕ], and q=cos[μ(ωn/ωa)cosϕ]. The terms fn and gn are cosine expansion coefficients for initial values, namely,
{F(x1)G(x1)}{T(x1)h(x1)}|t1=0={Tn(0)hn(0)}cos(λnx1)n=0{fngn}cos(λnx1).
Note that to investigate the phase locking phenomenon, one should remove the significant annual cycle of SSTA. Therefore, the solution for Eq. (10) is eventually written as
T(x1,t1)=n=1Cnsin[ωnt1+μωnωacos(ωat1ϕ)+θn]×cos(λnx1).
It is obvious that the solution reduces to the regular harmonic oscillation when μ = 0. Transforming to the (x, t) coordinate system, Eq. (26) becomes
T(x,t)=n=1Cnsin[ωnt+μωnωacos(ωatϕ)+θn]×cosλn(xu¯t).
Since cosλn(xu¯t)=cosλnxcosu¯t+sinλnxsinu¯t, Eq. (27) can further be written as
T(x,t)=n=1Cnsin[ωnt+μωnωacos(ωatϕ)+θn]cosu¯tcosλnx+n=1Cnsin[ωnt+μωnωacos(ωatϕ)+θn]×sinu¯tsinλnx.

c. The dynamics of phase locking

The phase locking of ENSO to the annual cycle means that the peak SSTAs tend to occur in December or January, which can be described by the zero points of the first derivatives of SSTAs, that is,
dTndt1=Cncos[ωnt1+μωnωacos(ωat1ϕ)+θn]A(t1)=0.
Since Cn is not always equal to zero, Eq. (29) requires
ωnt1+μωnωacos(ωat1ϕ)+θn=(m+12)π,
where m = 0, 1, 2, …. Equation (30) is a transcendental equation and is impossible to obtain its analytic solution. However, it is possible to conduct qualitative analysis. Let us first discuss a simpler case where μ = 0 so that Eq. (30) reduces to
ωnt1+θn=(m+12)π,
Its solution (e.g., labeled as t¯) satisfies
t¯=12[(m+12)θnπ]Pn,
where Pn=2π/ωn is the natural period(s) associated with the natural frequency ωn. To identify which calendar month t¯ belongs to, let us rearrange Pn to
Pn=12k+Pnm,
where k is an arbitrary nonnegative integer and Pnm is the remainder, which satisfies 0 ≤ Pnm ≤ 11 and can represent the calendar month. Note that here the time unit is set to month for convenience. Now Eq. (32) may be rearranged as
t¯=12[(m+12)θnπ]12k+12[(m+12)θnπ]Pnm.
According to Eq. (34), t¯ can be evenly distributed throughout any a calendar month with varying m. Therefore, t¯ is a solution with no phase locking.
Now let us turn around to identify whether Eq. (30) can represent any phase locking feature. Following Eqs. (31) and (32), its solution, labeled as t*, can be set to
t*=t¯+tp,
where tp denotes a small departure. Substituting Eq. (35) into Eq. (30), it is easy to derive
θp=μcos(ωat¯θp)+ϕ,
where θp = −(ωatpϕ). Equation (36) is also hard to solve. However, as shown in Fig. 1, its solution is the intersection of a straight line (θp) and a curve [μcos(ωat¯θp)+ϕ] if θp is treated as a variable. It is obvious that the solution is limited to a range
ϕμθpϕ+μ.
On the other hand, since θp is the phase of the cosine curve [μcos(ωat¯θp)+ϕ], it means when t¯ evenly appears in each calendar month, its phase θp is limited to a range centered on the phase ϕ of the annual cycle, that is, with the phase θp being phase locked to the phase of the annual cycle.
Fig. 1.
Fig. 1.

Schematic diagram of the solution to Eq. (36).

Citation: Journal of Climate 37, 8; 10.1175/JCLI-D-23-0527.1

To further illustrate the phase locking feature, Eq. (36) is rewritten as
tp=μωacos(ωat*ϕ).
It is easy to derive
tp>0,if12π<ωat*ϕ<32π
and
tp<0,if32π<ωat*ϕ<52π.
Now substituting the specific values of ωa=2π/12 and ϕ=(1/3)π (identified in section 3) into Eq. (39), it becomes
1012π=12π+13π<2π12(t¯+tp)<32π+13π=2212π
or
5<t¯+tp<11.
Note that the numbers 5 and 11 in Eq. (42) denote May and November, respectively. As previously pointed out, t¯ can be evenly distributed in any a calendar month; for example, t¯ is just located in July (t¯=6). Then the small positive departure tp will make t¯ move toward November, the phase of the annual cycle. This is true when t¯ is located from May to November. Similarly, Eq. (40) is reduced to
11<t*<17,
or in calendar month
1<t¯+tp<4,or11<t¯+tp<12,
where the numbers 1, 4, 11, and 12 denote January, April, November, and December, respectively. This demonstrates that when t¯ is distributed from November to next April, the small negative tp will also pull it back to close to November, the phase of the annual cycle. Besides, tp is proportional to μ, the amplitude of the annual cycle. This means the stronger the amplitude of the annual cycle is, the larger range tp can approach to, and the much closer to the phase of the annual cycle the solution t* will tend to and hence the more significant the phase locking feature will be.

Now, it is clear that the peak time of SSTA can be divided into two parts. The first part, labeled as t¯, can be evenly distributed in any a calendar month, representing the solution with no phase locking characteristics. The second part, labeled as tp, is limited in a small range that depends on the amplitude and phase of the annual cycle, representing the phase locking feature. When t¯ is located left (right) to the phase of the annual cycle, tp is a positive (negative) value to make sure the solution t* moves right (left) toward the phase of the annual cycle. Besides, when the amplitude of the annual cycle is stronger, the value of tp can realized a larger modulation to t¯, pulling it closer to the phase of the annual cycle. Particularly, if no annual cycle is introduced, tp = 0 and there will be no phase locking phenomenon. Finally, tp is independent of the spatial eigenmodes, which indicates that the phase locking characteristics will also be independent to the spatial eigenmodes.

3. The annual cycle of the basic-state thermocline depth

According to the previous investigation, the annual variation of the basic-state thermocline depth may play an important role in the phase locking of ENSO to the annual cycle (An and Wang 2001). On the other hand, the seasonable variation of the basic-state thermocline depth has been incorporated into the STO model and the analytic solution has been successfully obtained in section 2. To accurately describe the annual cycle of the basic-state thermocline depth, the seasonal departure of the equatorial (5°S–5°N mean) Pacific mixed layer depth from its annual mean in the simple ocean data assimilation (SODA) version 3.4.2 (Carton et al. 2018) is portrayed in Fig. 2a. Significant seasonal variation can be observed especially in the central eastern Pacific (around 180°–90°W) where the mixed layer depth anomaly decreases to a negative maximum value around April and then rebounds to positive maximum value around August and then declines again to form an obvious annual cycle. Furthermore, this significant annual cycle is also modulated by a semiannual variation, which is also significant in the second half of the year. When the positive maximum value begins to decline from around August, it just decreases moderately in October and November, keeping its positive anomaly, and then rebounds again in December and January. This semiannual cycle is most significant in the western equatorial Pacific. The semiannual variation may be associated with the direct solar radiation since the sun moves across the equator twice a year (Chen and Jin 2018).

Fig. 2.
Fig. 2.

(a) Seasonal variation of monthly mixed layer depth anomaly and (b) monthly sea surface height anomaly from their annual mean climatology. The contour level is 10 m for (a) and 10 mm for (b).

Citation: Journal of Climate 37, 8; 10.1175/JCLI-D-23-0527.1

Although it is not difficult to depict this semiannual variation of the mean mixed layer depth or the mean thermocline depth, it may interfere with the phase locking to the annual cycle. Therefore, it is better to focus on the annual cycle itself. Considering the fact that the sea surface height is a close proxy of the local thermocline depth (Shi et al. 2020), it is reasonable to replace the thermocline depth with the sea surface height which only has significant annual but no semiannual variation (Fig. 2b). The annual variations of the sea surface height and thermocline depth are quite similar although the phase of the thermocline depth seems to lead that of the sea surface height around one month (An and Wang 2001).

The annual variation can be directly seen from the zeroth-order cosine expansion coefficients for the monthly mean climatological sea surface height (dashed curve with square in Fig. 3). It forms a quite standard sine curve with a period of 12 months and with a phase ϕ=(1/3)π (solid curve in Fig. 3). On the contrary, the annual signal in the monthly climatological mean mixed layer depth (solid curve with asterisk in Fig. 3) is obviously modulated by the semiannual signal as previously discussed. Therefore, the annual cycle of the mean thermocline depth in this paper is replaced by the annual cycle of its close proxy, that is, the mean sea surface height.

Fig. 3.
Fig. 3.

Zeroth-order cosine expansion coefficients for the monthly mean climatological mixed layer depth (solid curve with asterisks), sea surface height (dashed curve with squares), and a standard sine curve (solid curve) to denote the annual cycle. To facilitate comparing, all curves are normalized.

Citation: Journal of Climate 37, 8; 10.1175/JCLI-D-23-0527.1

4. The features of ENSO phase locking

As discussed in section 2, the variations of SSTA are decomposed into the sum of a series spatial eigenmode which oscillates with their natural frequency. In addition, the phase locking is independent of spatial eigenmodes. To elucidate the characteristics of phase locking, the first two spatial eigenmodes that can represent the quasi-quadrennial (QQ) mode and the quasi-biennial (QB) modes as Li (2024) suggested are examined. The parameters values are set to u¯=0.1ms1, Kh0 = 1.0 × 10−8 K m−1 s−1, b = 1.25 × 10−5 m2 s−1 K−1, H0 = 50 m, c0=Kh0H0b=0.25ms1, and L = 1.8 × 107 m. The initial phases that are determined by initial conditions are set to 0 (θ1,2 = 0). The first two natural frequencies are calculated as ω1=(2π/55)month1 and ω2=(1/2)ω1. Modulated by the annual cycle [ωa=(2π/12)month1], the natural frequencies become 2π/(55×12)=(2π/660)month1 and (2π/330)month1, respectively. The original natural periods (55 and 27.5 months) are still significant but the oscillation shapes in each period have a little difference (Fig. 4). To demonstrate the phase preference, the amplitude and the phase of the annual cycle are set to μ=1/2 and ϕ=(1/3)π, and the initial phase varies from 0 to 2π (0 ≤ θ1,2 ≤ 2π) to calculate the histogram of the peak times for each spatial eigenmode. The time for each calculation is 1000 years with 5-day time step. Monthly averaged results are applied to identify the peak calendar months. Following Chen and Jin (2020), the El Niño (La Niña) events are defined as happening when the normalized time series exceeds one positive (negative) standard deviation.

Fig. 4.
Fig. 4.

Normalized temporal evolution for the (a) first and (b) second spatial eigenmodes.

Citation: Journal of Climate 37, 8; 10.1175/JCLI-D-23-0527.1

As shown in Fig. 5, the peak times of both El Niño and La Niña tend to occur toward the end the calendar year with the maximum probability time appears in November. Note that the peak time is consistent with that of the annual cycle. It is interesting to point out that the histogram of El Niño and La Niña for the time series of the second spatial eigenmode are totally the same as that of the first spatial mode. Actually, this is true for any a spatial mode (figures omitted). Therefore, the phase preference phenomenon is independent of the natural spatial eigenmodes. This further confirms the theoretical analysis in section 2. Comparing with observations (Chen and Jin 2020) that show the maximum probability of ENSO peak time appears in December for El Niño but in November for La Niña, the analytic results are a month earlier for El Niño and are simultaneous for La Niña. This further demonstrates that the STO model with annual varying parameters does have the capacity to reproduce the phase locking phenomenon.

Fig. 5.
Fig. 5.

Histogram of El Niño (red bars) and La Niña (blue bars) peak months for the time series of the first spatial eigenmode.

Citation: Journal of Climate 37, 8; 10.1175/JCLI-D-23-0527.1

According to Li (2024), the EP and CP El Niño events can be featured by the combinations of the first two spatial eigenmodes with different weights. For the EP (CP) Niño events, the weights for the two modes are set to 2:1 (1:2). As shown in Fig. 6, although the spatial and temporal variations present certain different if the first two modes are superposed with different initial phases, they basically reproduce the typical characteristics of the EP Niño events, that is, the warm SSTA appearing in the eastern and central equatorial Pacific. The situation is quite similar for CP Niño events (Fig. 7). It is interesting to point out that although the first two modes are superposed weight different weights and different initial phases, the calculated Niño-3.4 index time series present the same phase locking feature as Fig. 5 shows. It is easy to understand since the time series of each spatial mode obeys the same laws for phase locking. Note that Figs. 6 and 7 seem to mean that the La Niña events can also be classified into two types, although the existence of which is still controversial (Song et al. 2017).

Fig. 6.
Fig. 6.

Evolution of the prototypical EP El Niño events that are calculated by the combination of the first and the second spatial eigenmodes, the phases of which are different but the weights of which are the same.

Citation: Journal of Climate 37, 8; 10.1175/JCLI-D-23-0527.1

Fig. 7.
Fig. 7.

As in Fig. 6, but for the prototypical CP El Niño events.

Citation: Journal of Climate 37, 8; 10.1175/JCLI-D-23-0527.1

5. The importance of the annual cycle

The effects of the amplitude and phase of the annual cycle on the phase locking feature are analyzed in this section. It is obvious that there will be no phase locking phenomenon if there is no annual cycle (μ = 0). As shown in Fig. 8, the peak time of ENSO events evenly appearst in each calendar month with an equal probability of around 8.3%. With strengthening amplitude of the annual cycle (μ is increasing from 0 to 1), the phase locking phenomenon becomes more and more significant (Fig. 9). The overall probability of peak time that appears from August to the next February increases while the overall probability of appearing from February to August decreases (Fig. 9a). Specifically, the probability that the peak time appears in November basically linearly doubles from around 8% to exceeding 16% (Fig. 9b).

Fig. 8.
Fig. 8.

Histogram of El Niño (red bars) and La Niña (blue bars) peak months when there is no annual cycle.

Citation: Journal of Climate 37, 8; 10.1175/JCLI-D-23-0527.1

Fig. 9.
Fig. 9.

(a) Variation of the probability of El Niño or La Niña peak months and (b) variation of the probability when the El Niño or La Niña peak times occur in November with the amplitude of the annual cycle.

Citation: Journal of Climate 37, 8; 10.1175/JCLI-D-23-0527.1

The phase of the annual cycle does not modulate the amplitude of the phase locking feature but directly determines which calendar month the phase locking occurs. The peak time is basically phase locked to September to October when the phase of the annual cycle is zero and to November to the next January when the phase increases to (1/2)π, and to February to April when the phase continues to increase to π, and further to May to July when the phase continues to increase to (3/2)π, to eventually form an annual cycle (Fig. 10a). This can be directly seen if simply portraying the probability variations in November (Fig. 10b).

Fig. 10.
Fig. 10.

As in Fig. 9, but for the variations with the phase of the annual cycle.

Citation: Journal of Climate 37, 8; 10.1175/JCLI-D-23-0527.1

As theoretically analyzed in section 2, the phase locking feature is caused by tp, which is governed by Eq. (36) or Eq. (38) and is hard to obtain the analytic solution. Although Fig. 1 has vividly exhibited the solution, it only provides certain qualitative conclusions, such as the upper and lower limits of tp and its signs. Here the numerical solution is calculated to further concretize the theoretical analysis. According to Eq. (37), the stronger the amplitude of the annual cycle is, the larger range tp can approach to. However, if multiplied by ωa/μ, the variation range of tp would be normalized to [−1, 1]. It is interesting to note that the normalized tp obeys the same distribution, which says that there is the largest probability when tp is close to its upper and lower limits while the smallest probability when tp is close to zero (Fig. 11). When tp is added onto the no phase locking solution t¯, the final solution t* concentrated toward the phase of the annual cycle regardless of the calendar month in which t¯ is located (Fig. 12). Besides, the farther t¯ is located away from the phase of the annual cycle, the larger tp and its modulation effect is. Finally, the variation range of tp increases with the amplitude of the annual cycle. This means the modulation of tp becomes more significant to realize a more significant phase locking feature. Therefore, the calculation results further support the theoretical analysis in section 2.

Fig. 11.
Fig. 11.

Histogram of the dimensionless tp.

Citation: Journal of Climate 37, 8; 10.1175/JCLI-D-23-0527.1

Fig. 12.
Fig. 12.

Evolution of the solution t* with the amplitude of the annual cycle. When there is no annual cycle (μ = 0), the solution t* equals t¯ and is evenly distributed in a year. Note that solution t* can have a larger movement toward the phase of the annual cycle (November) with stronger amplitude of the annual cycle.

Citation: Journal of Climate 37, 8; 10.1175/JCLI-D-23-0527.1

6. Summary and discussion

This paper investigated the phase locking characteristics of ENSO by applying a newly developed STO model framework that decomposes SSTAs into a series of spatial modes, each of which oscillates with their natural frequency. It can be regarded as the spatial extension of the classic RO model. The STO model is derived from the thermodynamics and thermocline dynamic equations in the tropical Pacific and therefore has a solid physical foundation. As previous investigations suggested, the annual variations of parameters/background variables may play important roles in controlling the phase locking feature. Therefore, the annual cycle of the thermocline depth is introduced to the STO model. The analytic solution of the derived parametric STO model is a sine curve of a new variable combined by the natural frequencies and the introduced annual cycle. The time of the phase locking is then defined by the solving the peak times of the SSTA time series, that is, the zero points of the first derivative of SSTA time series. It can be further theoretically proven that the peaks of SSTA time series can be divided into two parts. The first part, representing the solution with no annual cycle, can be evenly distributed in any a calendar month. It is the solution with no phase locking feature. The second part, directly associated with the annual cycle, adds an increment onto the first part so that their sum, the zero points of the first derivative of SSTA time series, moves toward the phase of the annual cycle, that is, the occurrence of the phase locking feature.

With observed annual cycle of the thermocline depth that declines from January to May to approach the minimum value and then rebounds to maximum value in November, the results show that both El Niño and La Niña events tend to occur in November with maximum probability of around 12%. The probability decreases when the calendar months move far away from November, the phase of the annual cycle. It is just the phase locking characteristics and quite consistent with the observed phenomenon. Since the annual cycle is independent of the spatial modes, the phase locking feature can be found the same in any a spatial mode or their combinations. The strength of the phase locking feature basically monotonously increases with the amplitude of the annual cycle as a straight line. When the amplitude of the annual cycle approaches to its maximum reasonable value, the probability that the peak time happens in November can exceed 16% while when there is no annual cycle, the probability that the peak time happens in November is around 8%, equal to other calendar months. The calendar month of the peak time is only determined by the phase of the annual cycle. The calculation results further suggest that the phase locking solution (the second part of the solution) distributes with largest probability when it is close to its upper and lower limits that are jointly determined by the annual frequency and amplitude of the annual cycle. This distribution can make it insert maximum modulation to the no phase locking solution that is far away from the phase of the annual cycle to realize the concentration of the solution toward the phase of the annual cycle.

According to previous investigations, the annual cycle of the climate background may play an important role in the formation of the ENSO phase locking phenomenon. Different studies highlighted different factors, such as the annual cycle of the ENSO growth/decay rate (Chen and Jin 2020) and the seasonal variation of the mean thermocline depth (An and Wang 2001). In the present investigation, the results also stress the critical role in the annual cycle of the thermocline depth. It seems to support one of these factors may be the most important factor for modulating the ENSO phase locking. However, both theoretical analysis and numerical calculation imply that it is the annual cycle of the controlling parameters, rather than the parameters themselves, that determines the ENSO phase locking phenomenon. In other words, the annual cycle of any a parameter in the system may lead to the ENSO phase locking. This is reasonable since previous investigations had successfully reproduced the ENSO phase locking from the annual cycle of the different parameters. Considering the fact that the STO model can be further developed by introducing more complex physics, such as the seasonal growth/decay rate, the stochastic forcing and so on, the understanding for the ENSO phase locking may be further deepened with the continuous development of the STO model.

Acknowledgments.

This study was jointly funded by the National Natural Science Foundation of China (Grant 42275051) and the Fundamental Research Funds for the Central Universities. I wish to thank the two anonymous reviewers for their constructive comments.

Data availability statement.

The SODA version 3.4.2 data used in this study are openly available from the University of Maryland at https://dsrs.atmos.umd.edu/DATA/soda3.4.2/.

REFERENCES

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    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Neelin, J. D., and F. Jin, 1993: Modes of interannual tropical ocean–atmosphere interaction—A unified view. Part II: Analytical results in the weak-coupling limit. J. Atmos. Sci., 50, 35043522, https://doi.org/10.1175/1520-0469(1993)050<3504:MOITOI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Neelin, J. D., and Coauthors, 1992: Tropical air-sea interaction in general circulation models. Climate Dyn., 7, 73104, https://doi.org/10.1007/BF00209610.

    • Search Google Scholar
    • Export Citation
  • Neelin, J. D., F. Jin, and H. Syu, 2000: Variations in ENSO phase locking. J. Climate, 13, 25702590, https://doi.org/10.1175/1520-0442(2000)013<2570:VIEPL>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Rasmusson, E. M., and T. H. Carpenter, 1982: Variations in tropical sea surface temperature and surface wind fields associated with the Southern Oscillation/El Niño. Mon. Wea. Rev., 110, 354384, https://doi.org/10.1175/1520-0493(1982)110<0354:VITSST>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Stein, K., N. Schneider, A. Timmermann, and F. Jin, 2010: Seasonal synchronization of ENSO events in a linear stochastic model. J. Climate, 23, 56295643, https://doi.org/10.1175/2010JCLI3292.1.

    • Search Google Scholar
    • Export Citation
  • Stein, K., A. Timmermann, N. Schneider, F. Jin, and M. F. Stuecker, 2014: ENSO seasonal synchronization theory. J. Climate, 27, 52855310, https://doi.org/10.1175/JCLI-D-13-00525.1.

    • Search Google Scholar
    • Export Citation
  • Thompson, C. J., and D. S. Battisti, 2000: A linear stochastic dynamical model of ENSO. Part I: Model development. J. Climate, 13, 28182832, https://doi.org/10.1175/1520-0442(2000)013<2818:ALSDMO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Tziperman, E., L. Stone, M. A. Cane, and H. Jarosh, 1994: El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator. Science, 264, 7274, https://doi.org/10.1126/science.264.5155.72.

    • Search Google Scholar
    • Export Citation
  • Tziperman, E., S. E. Zebiak, and M. A. Cane, 1997: Mechanisms of seasonal–ENSO interaction. J. Atmos. Sci., 54, 6171, https://doi.org/10.1175/1520-0469(1997)054<0061:MOSEI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Vecchi, G. A., and D. E. Harrison, 2006: The termination of the 1997–98 El Niño. Part I: Mechanisms of oceanic change. J. Climate, 19, 26332646, https://doi.org/10.1175/JCLI3776.1.

    • Search Google Scholar
    • Export Citation
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Save
  • An, S., and B. Wang, 2001: Mechanisms of locking of the El Niño and La Niña mature phases to boreal winter. J. Climate, 14, 21642176, https://doi.org/10.1175/1520-0442(2001)014<2164:MOLOTE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • An, S., and F. Jin, 2011: Linear solutions for the frequency and amplitude modulation of ENSO by the annual cycle. Tellus, 63A, 238243, https://doi.org/10.1111/j.1600-0870.2010.00482.x.

    • Search Google Scholar
    • Export Citation
  • Battisti, D. S., 1988: Dynamics and thermodynamics of a warming event in a coupled tropical atmosphere–ocean model. J. Atmos. Sci., 45, 28892919, https://doi.org/10.1175/1520-0469(1988)045<2889:DATOAW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Battisti, D. S., and A. C. Hirst, 1989: Interannual variability in a tropical atmosphere–ocean model: Influence of the basic state, ocean geometry and nonlinearity. J. Atmos. Sci., 46, 16871712, https://doi.org/10.1175/1520-0469(1989)046<1687:IVIATA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Cane, M. A., and S. E. Zebiak, 1985: A theory for El Niño and the Southern Oscillation. Science, 228, 10851087, https://doi.org/10.1126/science.228.4703.1085.

    • Search Google Scholar
    • Export Citation
  • Carton, J. A., G. A. Chepurin, and L. Chen, 2018: SODA3: A new ocean climate reanalysis. J. Climate, 31, 69676983, https://doi.org/10.1175/JCLI-D-18-0149.1.

    • Search Google Scholar
    • Export Citation
  • Chen, H., and F. Jin, 2020: Fundamental behavior of ENSO phase locking. J. Climate, 33, 19531968, https://doi.org/10.1175/JCLI-D-19-0264.1.

    • Search Google Scholar
    • Export Citation
  • Chen, H., and F. Jin, 2021: Simulations of ENSO phase-locking in CMIP5 and CMIP6. J. Climate, 34, 51355149, https://doi.org/10.1175/JCLI-D-20-0874.1.

    • Search Google Scholar
    • Export Citation
  • Chen, Y., and F. Jin, 2018: Dynamical diagnostics of the SST annual cycle in the eastern equatorial Pacific: Part I a linear coupled framework. Climate Dyn., 50, 18411862, https://doi.org/10.1007/s00382-017-3725-7.

    • Search Google Scholar
    • Export Citation
  • Dommenget, D., and Y. Yu, 2016: The seasonally changing cloud feedbacks contribution to the ENSO seasonal phase-locking. Climate Dyn., 47, 36613672, https://doi.org/10.1007/s00382-016-3034-6.

    • Search Google Scholar
    • Export Citation
  • Hirst, A. C., 1986: Unstable and damped equatorial modes in simple coupled ocean-atmosphere models. J. Atmos. Sci., 43, 606632, https://doi.org/10.1175/1520-0469(1986)043<0606:UADEMI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hirst, A. C., 1988: Slow instabilities in tropical ocean basin–global atmosphere models. J. Atmos. Sci., 45, 830852, https://doi.org/10.1175/1520-0469(1988)045<0830:SIITOB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Jin, F., and J. D. Neelin, 1993: Modes of interannual tropical ocean–atmosphere interaction—A unified view. Part I: Numerical results. J. Atmos. Sci., 50, 34773503, https://doi.org/10.1175/1520-0469(1993)050<3477:MOITOI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Jin, F., J. D. Neelin, and M. Ghil, 1994: El Niño on the devil’s staircase: Annual subharmonic steps to chaos. Science, 264, 7072, https://doi.org/10.1126/science.264.5155.70.

    • Search Google Scholar
    • Export Citation
  • Jin, F., J. D. Neelin, and M. Ghil, 1996: El Niño/Southern Oscillation and the annual cycle: Subharmonic frequency-locking and aperiodicity. Physica D, 98, 442465, https://doi.org/10.1016/0167-2789(96)00111-X.

    • Search Google Scholar
    • Export Citation
  • Jin, F., H.-C. Chen, S. Zhao, M. Hayashi, C. Karamperidou, M. F. Stuecker, R. Xie, and L. Geng, 2020: Simple ENSO models. El Niño Southern Oscillation in a Changing Climate, M. J. McPhaden, A. Santoso, and W. Cai, Eds., Amer. Geophys. Union, 119–151, https://doi.org/10.1002/9781119548164.ch6.

  • Kang, I., and S. An, 1998: Kelvin and Rossby wave contributions to the SST oscillation of ENSO. J. Climate, 11, 24612469, https://doi.org/10.1175/1520-0442(1998)011<2461:KARWCT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kang, I., S. An, and F. Jin, 2001: A systematic approximation of the SST anomaly equation for ENSO. J. Meteor. Soc. Japan, 79 (1), 110, https://doi.org/10.2151/jmsj.79.1.

    • Search Google Scholar
    • Export Citation
  • Li, Y. K., 2024: A spatiotemporal oscillator model for ENSO. Theor. Appl. Climatol., 155, 32813296, https://doi.org/10.1007/s00704-023-04809-2.

    • Search Google Scholar
    • Export Citation
  • Lu, Z., and Z. Liu, 2019: Orbital modulation of ENSO seasonal phase locking. Climate Dyn., 52, 43294350, https://doi.org/10.1007/s00382-018-4382-1.

    • Search Google Scholar
    • Export Citation
  • McPhaden, M. J., S. E. Zebiak, and M. H. Glantz, 2006: ENSO as an integrating concept in Earth science. Science, 314, 17401745, https://doi.org/10.1126/science.1132588.

    • Search Google Scholar
    • Export Citation
  • Neelin, J. D., and F. Jin, 1993: Modes of interannual tropical ocean–atmosphere interaction—A unified view. Part II: Analytical results in the weak-coupling limit. J. Atmos. Sci., 50, 35043522, https://doi.org/10.1175/1520-0469(1993)050<3504:MOITOI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Neelin, J. D., and Coauthors, 1992: Tropical air-sea interaction in general circulation models. Climate Dyn., 7, 73104, https://doi.org/10.1007/BF00209610.

    • Search Google Scholar
    • Export Citation
  • Neelin, J. D., F. Jin, and H. Syu, 2000: Variations in ENSO phase locking. J. Climate, 13, 25702590, https://doi.org/10.1175/1520-0442(2000)013<2570:VIEPL>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Rasmusson, E. M., and T. H. Carpenter, 1982: Variations in tropical sea surface temperature and surface wind fields associated with the Southern Oscillation/El Niño. Mon. Wea. Rev., 110, 354384, https://doi.org/10.1175/1520-0493(1982)110<0354:VITSST>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Shi, J., A. V. Fedorov, and S. Hu, 2020: A sea surface height perspective on El Niño diversity, ocean energetics, and energy damping rates. Geophys. Res. Lett., 47, e2019GL086742, https://doi.org/10.1029/2019GL086742.

    • Search Google Scholar
    • Export Citation
  • Song, L., S. Chen, W. Chen, and X. Chen, 2017: Distinct impacts of two types of La Niña events on Australian summer rainfall. Int. J. Climatol., 37, 25322544, https://doi.org/10.1002/joc.4863.

    • Search Google Scholar
    • Export Citation
  • Stein, K., N. Schneider, A. Timmermann, and F. Jin, 2010: Seasonal synchronization of ENSO events in a linear stochastic model. J. Climate, 23, 56295643, https://doi.org/10.1175/2010JCLI3292.1.

    • Search Google Scholar
    • Export Citation
  • Stein, K., A. Timmermann, N. Schneider, F. Jin, and M. F. Stuecker, 2014: ENSO seasonal synchronization theory. J. Climate, 27, 52855310, https://doi.org/10.1175/JCLI-D-13-00525.1.

    • Search Google Scholar
    • Export Citation
  • Thompson, C. J., and D. S. Battisti, 2000: A linear stochastic dynamical model of ENSO. Part I: Model development. J. Climate, 13, 28182832, https://doi.org/10.1175/1520-0442(2000)013<2818:ALSDMO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Tziperman, E., L. Stone, M. A. Cane, and H. Jarosh, 1994: El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator. Science, 264, 7274, https://doi.org/10.1126/science.264.5155.72.

    • Search Google Scholar
    • Export Citation
  • Tziperman, E., S. E. Zebiak, and M. A. Cane, 1997: Mechanisms of seasonal–ENSO interaction. J. Atmos. Sci., 54, 6171, https://doi.org/10.1175/1520-0469(1997)054<0061:MOSEI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Vecchi, G. A., and D. E. Harrison, 2006: The termination of the 1997–98 El Niño. Part I: Mechanisms of oceanic change. J. Climate, 19, 26332646, https://doi.org/10.1175/JCLI3776.1.

    • Search Google Scholar
    • Export Citation
  • Wengel, C., M. Latif, W. Park, J. Harlaß, and T. Bayr, 2018: Seasonal ENSO phase locking in the Kiel Climate Model: The importance of the equatorial cold sea surface temperature bias. Climate Dyn., 50, 901919, https://doi.org/10.1007/s00382-017-3648-3.

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  • Zebiak, S. E., and M. A. Cane, 1987: A model El Niño–Southern Oscillation. Mon. Wea. Rev., 115, 22622278, https://doi.org/10.1175/1520-0493(1987)115<2262:AMENO>2.0.CO;2.

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  • Fig. 1.

    Schematic diagram of the solution to Eq. (36).

  • Fig. 2.

    (a) Seasonal variation of monthly mixed layer depth anomaly and (b) monthly sea surface height anomaly from their annual mean climatology. The contour level is 10 m for (a) and 10 mm for (b).

  • Fig. 3.

    Zeroth-order cosine expansion coefficients for the monthly mean climatological mixed layer depth (solid curve with asterisks), sea surface height (dashed curve with squares), and a standard sine curve (solid curve) to denote the annual cycle. To facilitate comparing, all curves are normalized.

  • Fig. 4.

    Normalized temporal evolution for the (a) first and (b) second spatial eigenmodes.

  • Fig. 5.

    Histogram of El Niño (red bars) and La Niña (blue bars) peak months for the time series of the first spatial eigenmode.

  • Fig. 6.

    Evolution of the prototypical EP El Niño events that are calculated by the combination of the first and the second spatial eigenmodes, the phases of which are different but the weights of which are the same.

  • Fig. 7.

    As in Fig. 6, but for the prototypical CP El Niño events.

  • Fig. 8.

    Histogram of El Niño (red bars) and La Niña (blue bars) peak months when there is no annual cycle.

  • Fig. 9.

    (a) Variation of the probability of El Niño or La Niña peak months and (b) variation of the probability when the El Niño or La Niña peak times occur in November with the amplitude of the annual cycle.

  • Fig. 10.

    As in Fig. 9, but for the variations with the phase of the annual cycle.

  • Fig. 11.

    Histogram of the dimensionless tp.

  • Fig. 12.

    Evolution of the solution t* with the amplitude of the annual cycle. When there is no annual cycle (μ = 0), the solution t* equals t¯ and is evenly distributed in a year. Note that solution t* can have a larger movement toward the phase of the annual cycle (November) with stronger amplitude of the annual cycle.

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