## 1. Introduction

A distinct seasonal predictability barrier (PB) has long been noticed in the tropical Pacific sea surface temperature (SST) variability and the associated El Niño–Southern Oscillation (ENSO) variability in rainfall (Walker and Bliss 1932; Wright 1979) and sea level pressure (Troup 1965; Webster and Yang 1992). This seasonal PB is characterized by a band of maximum decline in monthly autocorrelation function (ACF), sometimes also referred to as a persistence barrier (Troup 1965; Torrence and Webster 1998), at the fixed phase of May–June, and can be seen in the seasonal ACF of SST variability over the central–eastern Pacific (Fig. 1a; 95% significance level is 0.3 for 46 years of data) and its lag gradient (Fig. 1b) (e.g., Ren et al. 2016). This seasonal phase-locking of PB implies that, regardless of the initial month, a damped persistence forecast loses its predictability most rapidly in the following May, forming the so-called spring PB of ENSO. This barrier in persistence forecast also corresponds to a barrier of predictability in ensemble forecast (e.g., Levine and McPhaden 2015). Since the PB season occurs close to the time of minimum SST variance, it has often been interpreted as caused by the minimum SST variance in spring (left panel of Fig. 1a), when the weak ENSO signal renders it most vulnerable to noise forcing and, in turn, a transition from one climate state to another (e.g., Webster and Yang 1992; Xue et al. 1994; Torrence and Webster 1998; Boschat et al. 2013; Levine and McPhaden 2015; Tasambay-Salazar et al. 2015; Ren et al. 2016; Moon and Wettlaufer 2017). Recent studies have also found similar seasonal PB in other regions, for example, the western North Pacific in the so-called reemergence region (38°–42°N, 160°E–180°; Alexander et al. 1999), where SST variability also exhibits a band of maximum ACF decline phase-locked to June, or the so-called summer PB in the North Pacific (Figs. 1c,d) (Zhao et al. 2012). Since the SST variability in the North Pacific and tropical Pacific is generated predominantly by different mechanisms (the former by climate noise, the latter by ocean–atmosphere feedback), it is interesting that a similar seasonal PB is generated in both regions, implying some general mechanism for the generation of seasonal PB. Meanwhile, in contrast to the tropical Pacific where the PB month is locked close to, and is therefore interpreted as generated by, the minimum SST variance, the summer PB in the North Pacific occurs between the minimum and maximum SST variances, the latter of which occurs in late summer due to the shallow mixed layer (e.g., Zhao et al. 2012). This suggests that the seasonal PB in the North Pacific, unlike the tropical Pacific, may not be generated simply by the minimum SST variance.

Here, we study the seasonal PB in the framework of the simplest stochastic climate model: the Langevin equation (Hasselmann 1976), with the incorporation of a seasonal cycle in the damping rate and noise forcing. In the discrete form, this seasonal stochastic model corresponds to the first-order autoregressive (AR1) model with periodic coefficients and the analysis in this system is referred to as the cyclostationary analysis (Bennett 1958; Hurd 1975). ENSO PB has been studied extensively in the discrete AR1 model (Torrence and Webster 1998; Ren et al. 2016), as well as the continuous models of the Langevin model (Moon and Wettlaufer 2017) and delayed oscillator models (Stein et al. 2010; Levine and McPhaden 2015). These studies have shown the existence of the seasonal PB and discussed its responses to some model parameters. Nevertheless, the solutions so far are either derived numerically or expressed in rather complex forms (e.g., Torrence and Webster 1998; Levine and McPhaden 2015) such that important questions on the seasonal SB in general still remain not fully understood. In particular, we are interested in three questions regarding the condition, timing, and intensity of the seasonal PB. First, what is the necessary forcing condition for a seasonal PB? Second, what determines the timing and intensity of a seasonal PB? Third, are there other regions in the World Ocean where seasonal PB is present and, if yes, why they are present?

This paper is an attempt to understand the general features of the seasonal PB with the focus on the three questions raised above. In particular, we will derive a general, albeit asymptotic, solution of the seasonal Langevin equation. This solution will enable us to develop a theory for the necessary forcing condition, the timing, and the intensity of the seasonal PB, shedding light on the mechanism of seasonal PB. We will also apply our theory to assess the seasonal PB in the SST variability observations over the world. We will show that seasonal PB is a general feature of a seasonal stochastic model once the seasonal forcing, either in growth rate or in noise forcing, exceeds a modest threshold. The timing of a PB depends on the damping rate of the system as well as the type and magnitude of the seasonal forcing, while the intensity of a PB depends on the magnitude of the seasonal forcing. Our theory is found to be able to explain some major features of the seasonal PBs in the observed SST variability over the world.

The paper is organized as follows. The seasonal stochastic climate model is introduced in section 2 and a general solution is derived for the necessary forcing condition, the timing, and the intensity of the seasonal PB. The seasonal PB is then studied in the idealized cases forced by the seasonal growth rate and noise forcing in sections 3 and 4, respectively. The theory is applied to the observed SST variability in the tropical and North Pacific in section 5 and for the World Ocean in section 6. Section 7 presents the conclusions and some discussions. In a companion paper (Y. Jin et al. 2018, unpublished manuscript), we study in detail another feature of the phase-locking: the phase-locking of the SST variance and its relation with the lag-1 persistence.

## 2. General solution for seasonal predictability barrier

*T*is the SST anomaly. The damping rate

*b*(

*t*), or growth rate −

*b*(

*t*), has an annual mean magnitude

*A*asThe stochastic forcing is a white noise of zero mean, with a variance of annual mean magnitude

*D*(<1):where the angle brackets denote the ensemble covariance. The phase of the seasonal cycle here is chosen such that month

*t*= 0 corresponds to the month of maximum growth rate −

*b,*and it leads the month of maximum noise intensity

*M*is the effective amplitude of the seasonal cycle of the combined growth rate and noise forcing. It reduces to the amplitude of the seasonal growth rate

*M = AB*in the absence of seasonal noise forcing (

*D*= 0) and reduces to the amplitude of the seasonal noise forcing

*M = D*in the absence of seasonal growth rate (

*A*= 0).

^{1}

^{2}Therefore, the necessary forcing condition for a maximum ACF decline, or PB, isThat is, a PB emerges when the seasonal cycle amplitude

*M*relative to the damping rate

*B*exceeds a threshold of 1/2. Once (2.19) is satisfied, the PB exists throughout the year and remains on the fixed phasewherewith

The PB solution above shows two interesting features. First, the forcing magnitude has to exceed a threshold to generate a PB, and second, the PB, once generated, occurs on the same calendar month, forming a seasonal PB. The forcing threshold (2.19) is not surprising, although, to our knowledge, a threshold has not been derived and discussed in the literature. Without a seasonal cycle (*M* = 0), the ACF *M*/*B* >1/2 as in (2.19)], obviously, it will be able to overwhelm the mean damping rate, and the change of ACF decline will no longer be monotonic. This generates a maximum ACF decline or PB. Second, and most interestingly, it is not obvious at all that the PB, once generated, has to be phase-locked to the same calendar month and then form a seasonal PB. Our analytical solution demonstrates, therefore, that the phase-locking of PB, or seasonal PB, is an intrinsic feature of the seasonal stochastic system: once maximum ACF decline is generated, it occurs along the same calendar month

*M*, while the first year intensity

*B*.

It should be noted that caution should be taken on the application of the leading-order solutions of the PB to more general cases, notably the forcing threshold (2.19), the PB month (2.22), and intensity (2.25). Formally, these asymptotic solutions are valid in the limit of weak damping *B* ≪ 1 in (2.15), which is sufficient to ensure *AB* ≪ 1 in (2.8) and *DB* ≪ 1 in (2.13). Nevertheless, there are also hints that these PB features may still be largely valid beyond the weak damping limit to the realistic range of *B*, up to ~1. First, these features can also be shown to be largely valid in an intermediate asymptotic solution (not shown), which uses the approximate ACF (2.14a) but calculates the gradients numerically, instead of using (2.16) and (2.17). This intermediate solution only requires *AB* ≪ 1 and *DB* ≪ 1, instead of *B* ≪ 1. Second, these features seem to be also reasonably valid in the full solution of the Langevin Eq. (2.1) that is derived all numerically (see appendix B; see also Figs. 2, 4, and 7).

## 3. Seasonal predictability barrier forced by seasonal growth rate

*A*> 0) in the absence of seasonal noise variation (

*D*= 0), such that

*M = AB*,

*B*because of the cancelation of two opposite effects of the damping rate on PB: a larger

*B*increases the damping, which is unfavorable for a PB, but also increases the amplitude of the seasonal cycle forcing (=

*AB*), which is favorable for a PB. Following (2.22), the PB month is reduced towhereThe month of minimum variance in (2.12) is reduced toThe variance now lags the growth rate by

*A*(in

*B*(in

Some general features of the PB month can be seen in three examples in the ACF maps in Fig. 2 (and more examples in Figs. B1a and B2a), which are obtained from the full solution (2.6) and (2.7) numerically, in the case of a weak damping *B* = 0.2 and in turn *A* = 0.3; Fig. 2a), no PB is generated and the magnitude of the ACF decline decreases monotonically with lag. When *A* increases above the threshold (*A* = 1), a PB emerges with the PB locked to month of *A* = 5 (Fig. 2c),

The general feature of the PB timing can be understood from two limiting cases as shown schematically in Figs. 3a and 3b. For a strong damping, the SST response to the growth rate is instantaneous, so that the variance is locked in phase with the growth rate such that *A* increases from the threshold amplitude 1/2 to *A* ≫ 1, according to (3.2) (Fig. 2a). Intuitively, the seasonal PB can be understood as caused by the rapid decline of the growth rate, which leads to the maximum decline in SST signal and, in turn, the ACF. Equation (3.5) shows that *B* ≪ 1; Fig. 2b), the SST response lags the growth rate forcing by a season *A* ≫ 1), (3.3) gives *B* ~ 1 (as shown in Figs. B1 and B2).

*B*≪ 1 from (2.25) asTherefore, the PB intensifies mainly with the amplitude of the seasonal cycle

*AB*. This can be seen in the more distinct PB bands in the examples shown in Figs. 2a–c, with increasing

*A*and, in turn,

*AB*. The increased PB intensity with

*A*can also be seen more clearly in the examples in Fig. B2b.

## 4. Seasonal predictability barrier forced by seasonal noise forcing

*D*> 0) in the absence of seasonal growth rate (

*A*= 0), such that, from (2.11), (2.14b), and (2.14c),where, for simplicity, we have set

*D*exceeds a small damping-dependent threshold

*B*/2. The PB month (2.23) is reduced towhereThe minimum variance month remains the same as (3.4) and therefore, again, in general, the timings of PB and minimum variance differ as

Some features of the PB month can be seen in three examples of weak damping rate (*B* = 0.2) in the full solution in Fig. 4 (also Figs. B1b and B2c). For a weak seasonal forcing (*D* = 0.1) that just exceeds the threshold (Fig. 4a), a PB emerges at the calendar month *D* increases to 0.3 and, furthermore, to 0.98 (Figs. 4b,c),

General features of the PB can be discussed schematically as in Figs. 3c and 3d. For strong damping, *D*/*B*, we have

*D*from Fig. 4a to Fig. 4c (also see examples in Fig. B2d). Some previous work on seasonal PB tend to use the SST variance, and the related signal-to-noise ratio, to explain the PB (e.g., Xue et al. 1994; Torrence and Webster 1998). In contrast, here, we interpret the PB, the SST variance as well as their relationships all as the direct response to the seasonal forcing in the unified framework of a linear stochastic climate model. Our study shows that the relation between PB month and minimum variance is different between the growth rate forcing and noise forcing.

In the examples in Figs. 2 and 4, we also plotted the minimum ACF, or minimum persistence (gray crosses). It is interesting that the minimum persistence differs dramatically from the seasonal PB. First, the minimum persistence is not phase-locked to the same calendar month, because the lag increases twice the rate of the initial forecast month. Second, the initial month of minimum persistence differs from the PB month, but approaches the PB month for weak damping and strong seasonal cycle forcing. Finally, the minimum persistence is always present, with no threshold on the forcing, even in the case of very weak seasonal growth rate in Fig. 2a. These features are also present in the observation in the tropical Pacific and North Pacific (Figs. 1a,c) and will be discussed in more details in appendix A.

## 5. Application to tropical Pacific and North Pacific SSTs

Now, we apply the theory to the observed SST variability in the tropical Pacific and North Pacific, which have been briefly discussed earlier in Fig. 1. The monthly SST data are the Hadley Centre Sea Ice and Sea Surface Temperature dataset (HadISST) from 1960 to 2005 (Rayner et al. 2003). The SST anomaly is derived after subtracting the climatological seasonal cycle and linear trend. The ACF is constructed using two models (see appendix C for details). The first is the discrete cyclostationary AR1 model (e.g., Hasselmann and Barnett 1981; Ruiz de Elvira and Lemke 1982; Torrence and Webster 1998), in which the lag-1 coefficient is proportional to the growth rate. The 12-month seasonal cycles of SST variance, growth rate, and noise variance are calculated from the observation and the seasonal ACF is calculated using the AR1 model solution (appendix C). The second model is the seasonal Langevin equation [(2.1)] with the model parameters estimated from the observational SST; the model solution is then obtained numerically in the time step of 1 month (appendix C). The two approaches are complementary. The AR1 model, which has been used extensively in the past, captures the full variability of the original SST time series, but cannot be assessed against our theory quantitatively. The stochastic climate model can be assessed against our theory quantitatively, but loses high harmonics of SST variance. Here, we reexamine the seasonal PB in the tropical Pacific and North Pacific shown in Fig. 1, leaving the discussion to the World Ocean to the next section.

The tropical Pacific has a weak damping (*B* ~ 0.2, corresponding to *A* = 4, brown curve in the left panels of Figs. 5a,d) and the growth rate becomes unstable from June to November with the peak in September, consistent with previous works (e.g., Moon and Wettlaufer 2017). Physically, the seasonal cycle of the growth rate can be understood from the coupled ocean–atmosphere instability in terms of the BJ index (Jin et al. 2006). Here, however, we only focus on the generation mechanism of the spring PB. The spring PB is forced predominantly by the growth rate. This can be seen in sensitivity experiments forced by the seasonal growth rate (Figs. 5a,d), noise forcing (Figs. 5b,e), and combined growth rate and noise forcing (Figs. 5c,f) in both the AR1 (Figs. 5a–c) and stochastic climate (Figs. 5d–f) models (see appendix C for details). A comparison of the SST variance (black curve in the left panel of each figure) of different forcing scenarios shows clearly the overwhelming role of the seasonal growth rate forcing, consistent with previous works (e.g., Torrence and Webster 1998; Levine and McPhaden 2015). The seasonal growth rate also forces the distinct spring PB (black dots), as seen in the similar ACF forced by the growth rate (Figs. 5a,d) and the combined forcing (Figs. 5c,f) and the observation (Fig. 1a). This PB is caused by the large amplitude of seasonal cycle *A* = 4, which well exceeds the forcing threshold of 1/2 as shown in (3.1). The PB month leads the minimum variance slightly, both lagging the minimum growth rate by about a season, consistent with the theory in the limit of weak damping and strong seasonal cycle (Fig. 3b). This coincidence of the timings of the PB and minimum variance has been interpreted as the weak ENSO signal rendering itself most vulnerable to noise forcing (e.g., Webster and Yang 1992; Xue et al. 1994; Torrence and Webster 1998; Levine and McPhaden 2015). However, as discussed in section 3, the PB month does not have to be locked with the minimum variance in general. Instead, our stochastic climate model offers an alternative explanation. The spring PB of ENSO is forced by the rapid decline of the growth rate late in the fall, which forces a rapid decline of the SST signal and, in turn, its persistence; the persistence decline is then delayed by the weakly damped response and the strong seasonal cycle, such that the PB month approaches the month of minimum variance, forming the spring PB. The PB month coincides with the minimum variance here because of the weak damping and strong seasonal growth rate.

It should be noted that both models, although capturing the PB timing in the observation, underestimate the PB intensity substantially (Figs. 5c,f vs Fig. 1a). It has been suggested that the intensity of the spring PB of ENSO can be enhanced by state-dependent noise. The main source of the noise is the wind bursts in the western to central equatorial Pacific, which may act as state-dependent noise for ENSO because their generation is enhanced by increased SSTs and associated convection (Levine and McPhaden 2015). Here, this underestimation is also contributed significantly by the deficiency of the AR1 model and the Langevin model, which fail to represent the oscillatory feature of ENSO, a point to be returned to later.

In contrast to ENSO, the summer PB in the North Pacific SST is forced by the seasonal noise forcing (Fig. 6). This is not surprising because the seasonal growth rate is much weaker (*A* = 0.3), and the seasonal noise forcing is much stronger (*D* = 0.91) than in the tropical Pacific. As a result, both the SST variance and ACF forced by the noise forcing (Figs. 6b,e) almost reproduce those by the combined forcing (Figs. 6c,f) and in the observation (Fig. 1c). Now, the damping rate is somewhat strong *B* = 0.8 (corresponding to*D* in summer. Since the noise forcing (2.3) in our model (2.1) represents the effective noise forcing, which is proportional to the atmospheric noise forcing divided by the heat capacity of the ocean. Physically, this maximum *D* in summer is caused mainly by the shallower mixed layer, instead of the atmospheric variability itself (Zhao et al. 2012). Therefore, the summer PB in the western North Pacific can be understood as forced first by the declining noise forcing from summer toward the fall and then a delayed response time. The seasonal growth rate is too weak to force a PB by itself (Figs. 6a,d), because *A* = 0.3 is below the forcing threshold 1/2. Overall, the major features of the seasonal PB are consistent with our theory in both the tropical and North Pacific.

## 6. Application to the World Ocean

We now extend the application of our theory to the SST variability over the world. We will compare five analyses of seasonal PB, in which the ACFs are constructed by five methods: the direct observation calculation, the AR1 model solution, the AR2 model simulation, and the full and approximate solutions of the seasonal Langevin equation (appendix C). The direct observational analysis provides the benchmark that includes the full variability and dynamics. The AR1 model is used as the simplest linear stochastic model to capture the full variability, while the AR2 model is a further improvement of the AR1 model with the inclusion of oscillatory variability. The full and approximate solutions of the Langevin equation are to test our theory against the observation for the annual harmonic variance.

*t*, we identify the lag of PB

To assess the PB over the World Ocean, we calculate in Fig. 7 the PB intensity

Major features of the seasonal PB over the world show strong similarity between the observation (Figs. 7a–c) and the AR1 model (Figs. 7g–i). In both analyses, first, the region of the sharpest seasonal PB occurs in the tropical Pacific (around the ENSO region marked in Fig. 7a), with the entire region of *σ*_{B} < 1–2 (Figs. 7c,i). The PB month (Figs. 7b,h) occurs from April to July in the equatorial region, consistent with the spring PB of ENSO discussed earlier, and shifts to earlier spring south of the equator. Second, the midlatitude North Pacific (around the NP region marked in Fig. 7a) also exhibits seasonal PB with

In comparison with the observation, the AR1 model also shows a PB intensity of a similar spatial pattern, but with an overall weaker magnitude (within the dotted regions; cf. Figs. 7g and 7a). This underestimation of PB intensity has been seen in the ENSO region discussed earlier in Fig. 5. One reason of the weaker PB in the AR1 model is its inability to simulate the oscillatory evolution features in the observation. This is confirmed by the AR2 model simulation, which indeed shows a much enhanced PB intensity (Fig. 7d), now comparable with the observation (Fig. 7a), while the standard deviation and mean of the PB month (Figs. 7e,f) remain similar to the observation and the AR1 model.

The major features of the seasonal PB discussed above are also captured by the full solution of the stochastic climate model (Figs. 7j–l), in both the PB region (of small *B*, lowering the forcing threshold for the noise forcing, as shown in (4.1). These higher harmonics can therefore distort the seasonal PB established by the annual harmonics in the AR models. Furthermore, the lack of higher harmonics also leads to an underestimation of the PB intensity in the theory (Fig. 7j) than in the AR1 solution (Fig. 7g). In the meantime, the overall larger

Finally, major PB features identified in the full solution of the Langevin equation (Figs. 7j–l) are well approximated in the asymptotic solution (Figs. 7m–o). The PB intensity and month are almost identical (Figs. 7j,k vs Figs. 7m,n). The PB regions are also very consistent in the maps of larger *B* is large. This underestimation is caused by the overestimation of the forcing threshold (4.1) in the asymptotic solution under the seasonal noise forcing, as discussed in appendix B (see Fig. B1b). Overall, however, in spite of the formal assumptions of weak seasonal cycle in (2.8) and (2.13) and the weak damping in (2.15), the asymptotic solution is able to capture the major features of the full solution in the realistic regime of model parameters.

Given the success of the asymptotic solution, we can interpret the seasonal PB in the observation using the asymptotic solution in terms of the damping rate and seasonal forcing as shown in Fig. 8. Figure 8e shows the theoretical forcing threshold *M*/2*B* in (2.19). It is seen that the regions of seasonal PB discussed before, such as the tropical Pacific, midlatitude and western tropical North Pacific, and the Southern Ocean, are indeed consistent with the threshold of *M*/2*B* > 1. Furthermore, this threshold is reached mainly by the large seasonal cycle amplitude *M*. The large amplitude is contributed predominantly by the seasonal growth rate *A* > 1/2 in the tropical Pacific (Fig. 8b) but the seasonal noise forcing *D > B*/2 in the extratropics (cf. Figs. 8c and 8a).

We now discuss six regional PBs in more details as for the tropical and North Pacific in section 5. The domains and the corresponding model parameters of these regions, along with the tropical and North Pacific regions, are listed in Table 1, with domains also marked in Figs. 7a and 8e. The model parameters and the presence of PB for each region are also summarized in appendix B (Fig. B1, cyan circles). The western tropical North Pacific (WP; 10°–15°N, 130°–160°E) shows a sharp PB in June in the observation

Model parameters for the eight regions. The 95% significance levels are shown in parentheses for *A*, *B*, and *D*. The significance level is calculated with the Monte Carlo method with 1000 simulations, each using a random Gaussian error of the same variance replacing the residual of the annual harmonic fitting.

There are also regions of large SST variance but no clear seasonal PB. This can be seen in the ACF of the Kuroshio (KS; 22°–36°N, 122°–150°E) *A* > 1/2) to generate a PB (in April), the noise forcing alone is too weak to overcome the strong damping (*D* < *B*/2) to generate a seasonal PB, and the combined forcing forces no PB, suggesting the dominant role of the noise forcing. Yet, in the AR1 solution, seasonal growth rate and the combined forcing both force a PB (in April; not shown) whereas the seasonal noise forcing forces no PB, suggesting a dominant role of growth rate forcing. The difference between the two models is likely caused by the higher harmonics, which are strong as seen in the growth rate and variance (left panel of Fig. 9d). In the GS region *β* ~ 90° and therefore the two forcing amplitudes tend to cancel each other on the combined forcing amplitude *M* in (2.14c). In the AR1 model solutions to both KS and GS regions, the combined forcing is still able to force a spring PB. This also suggests that the absence of a clear PB in both regions in the observation (Figs. 9d,e) may be caused by sampling errors. The sampling error is particularly effective in distorting the PB because of the strong damping and in turn the short persistence in these regions.

Finally, there are also widespread regions completely absent of seasonal PB, even with large SST variance, such as the region of Malvinas Current (MC; 40°–50°S, 60°–30°W). There is no seasonal PB in the ACF (Fig. 8f;

In sum, clear seasonal PB can be identified in various regions of the World Ocean; these seasonal PBs are caused mainly by the seasonal growth rate in the tropics but by the seasonal noise forcing in the extratropics. These seasonal PBs can largely be understood in terms of our theory as the forcing magnitude exceeding the threshold. In some extratropical regions where forcing parameters are near the margin of the threshold, the seasonal PB either is absent or is generated/distorted by more subtle factors, such as sampling error and higher harmonics.

## 7. Conclusions and discussion

### a. Conclusions

We have developed a theory for the seasonal PB in the simplest stochastic climate system in the Langevin equation. Our results can be summarized as addressing the three questions raised in the introduction.

First, what is the necessary forcing condition for a seasonal PB? A maximum decline of ACF, or PB, is generated when the magnitude of the seasonal forcing, in either growth rate or noise forcing, overwhelms the damping rate, such that the forcing/damping ratio exceeds a modest forcing threshold [*M*/*B* > 1/2 as in (2.19)]. Once generated, all the PBs are phase locked to the same calendar month [

Second, what determines the timing and intensity of a seasonal PB? The season of the PB [

Third, are there other regions in the World Ocean where seasonal PB is present and, if yes, why they are present? There are many regions where a seasonal PB is present, mainly because of the strong seasonal forcing. In the tropics, the seasonal PB tends to be forced by a strong seasonal growth rate, whereas in the extratropics the seasonal PB tends to be generated by a strong seasonal noise forcing. Consistent with the observation, our theory predicts seasonal PBs in some regions, such as the austral summer PB in the Southern Ocean and the spring PB in the western tropical Pacific.

Overall, our theory is able to predict most major features of the seasonal PB in the observation. Therefore, we suggest that this theory can serve as a null hypothesis for the seasonal PB of climate variability in general.

### b. Discussion

Our theory suggests that a seasonally phase-locked PB is an intrinsic feature of a stochastic climate system with a moderate seasonal forcing. Given the applicability of the stochastic climate model to a wide range of climate variability (Hasselmann 1976; Penland and Magorian 1993), our theory may have implications for seasonal to interannual climate variability in general. Although the seasonal PB has been studied most intensively on the tropical Pacific ENSO, it may be applied to other regions and for other climate variables that exhibit significant seasonal modulations, such as sea ice variability (Moon and Wettlaufer 2011) and the seasonal thermocline (Alexander et al. 1999). In practice, however, the study of seasonal PB is more interesting for the case of weak damping, or longer persistence, because a strongly damped system leads to a rapid decline of ACF and therefore little predictability after the decorrelation time. Furthermore, since our theory may also have implications to a general stochastic climate system forced by a periodic forcing. For example, diurnal cycle is the other strong periodic forcing on the atmosphere and our theory in principle also applies to short-term weather prediction with diurnal cycle. Does the short-term weather forecast exhibit a predictability barrier phased locked to the diurnal cycle?

Much further work is needed to further improve our PB theory. First, we have focused on the annual harmonic here, while the full seasonal cycle in the observation often includes significant subannual higher harmonics. Therefore, the impact of higher harmonics and their interaction with the annual harmonic need to be further studied. Second, our theory is developed for the Langevin equation or AR1 model, while climate variability in some regions, such as the tropical Pacific ENSO, also exhibits significant oscillatory features that are better described in an oscillatory model or an AR2 system. Our current results show empirically that the AR1 model here severely underestimates the intensity of the PB. Therefore, to better assess the PB intensity, the seasonal PB theory should be developed further in an AR2 model or an oscillatory model (e.g., Stein et al. 2010; Levine and McPhaden 2015) and even multidimensional system (e.g., Penland and Magorian 1993).

We thank four anonymous reviewers for their comments on several versions of this paper. This work is supported by Chinese Ministry of Science and Technology Grant 2017YFA0603801, Natural Science Foundation of China Grant 41630527, and U.S. National Science Foundation Grant AGS-1656907.

# APPENDIX A

## Minimum ACF

*t*, the lag of maximum ACF decline

*t*in (2.20); the lag of minimum ACF

*t*in (A.2), so that

*r*(

*t*,0) = 1 and the minimum ACF

# APPENDIX B

## Comparison of the Asymptotic Solution with the Full Solution

The asymptotic solutions for seasonal PB in (2.19) and (2.22) are derived in the regime of weak seasonal forcing and weak damping under the assumption of (2.8), (2.13), and (2.15). Yet, as in the cases of many other asymptotic solutions, this asymptotic solution holds reasonably well in a parameter regime broader than the formal assumption. Here, the asymptotic solution is largely valid up to *B* ~ 1. This can be seen by comparing the asymptotic solution with the full solution of (2.6) and (2.7) in Figs. B1a and B1b (black circles). The forcing threshold under the seasonal growth rate in (3.1) is compared with the full solution in Fig. B1a in a set of experiments of varying *B* and *A*. It is seen that the threshold amplitude *A* is generally indeed close to the threshold 1/2 derived from the asymptotic solution. When *B* increases beyond 1, the threshold amplitude starts to exceed 1/2 modestly. This suggests that the asymptotic threshold *A* = 1/2 is an underestimation of ~10%–20% for large values of *B*. Similarly, under the seasonal noise forcing, Fig. B1b shows that the threshold *D = B*/2 in (4.1) is also a good approximation to the full solution. Opposite to the growth rate case in Fig. B1a, however, at large *B*, the threshold becomes smaller than the asymptotic threshold, so the latter has a modest overestimation of 10%–20%.

The threshold can also be seen consistent with the eight regions discussed in Fig. 1 (ENSO and NP) and Fig. 9 (WP, SO, IDM, KS, GS, and MC), which are shown in cyan circles in Fig. B1a (Fig. B1b) when forced by the seasonal growth rate (noise forcing) alone. Solid and hollow circles represent the presence and absence of a seasonal PB, respectively. In addition, a red plus sign indicates the presence of seasonal PB when forced by the combined growth rate and noise forcing. In some regions, the seasonal PB forced by the combined forcing shows some difference from that under the single forcing because of the interference between the two forcing. For example, NP has a seasonal PB in Fig. B1a even though its seasonal growth rate *A* is well below the threshold; the IDM has a seasonal PB in Fig. B1b even though the seasonal noise forcing *D* is well below the threshold, because of the interference between the two forcing.

Figure B2 also shows that the PB month and intensity, as functions of the magnitude of seasonal forcing, compare well between the asymptotic solution and the full solution. Two cases are shown, one weak damping *B* = 0.2 (black) and the other strong damping *B* = 0.6 (red). For the growth rate forcing, under a weak damping, the seasonal PB emerges (black solid) only after *A* exceeds 0.6, consistent with the forcing threshold *A* = 1/2 (Fig. B2a). The PB month *A* ~ 0.5 to a saturation month at around month −4 toward *A* ~*~*2. This full solution is well approximated by the asymptotic solution (3.2) even for a large amplitude of *A* up to 6 (black dot). For very large *A*, the PB month saturates toward *A* > 0.6 and the PB month increasing with *A*, except now the PB month is shifted earlier by about a month (red solid). This overall change of the PB month is also approximately by the leading-order solution (red dot), although the error is somewhat larger, because of the larger *B*.

The total intensity of the PB in the full solution increases almost linearly with *A* in the full solution (black and red solid lines) (Fig. B2b). This change is approximated by the asymptotic solution (3.6) well for the case *B* = 0.2 (black dot) and also reasonably well for the case of *B* = 0.6 (red dot). Overall, all the major features of the seasonal PB are in good agreement with the asymptotic solution. The PB month and intensity of the seasonal PB in response to seasonal noise forcing can be discussed similarly as shown in Figs. B2c and B2d. Note that now the seasonal PB emerges at increasing values of *D* for increasing *B*, consistent with the asymptotic threshold *D* = *B*/2 in (4.1).

# APPENDIX C

## Methods for Constructing ACF from the Monthly Observation

Five methods are used for the construction of the ACFs for the analysis of seasonal PB from the monthly SST observation.

### a. Method 1: Direct observational calculation

*t*= 12

*n + m*is in month, with

*n*as the year and

*m*as the calendar month. The autocorrelation function (ACF) for calendar month

*m*and lag

*k*is calculated aswhere the covariance and variance for the calendar month

*m*arewith

*N*= 46 being the total years. All the calendar month indices here are periodic as

### b. Method 2: AR1 model solution

*m*is calculated asThe SST variance in the AR1 model forced by seasonal growth rate and noise forcing can be calculated as [(A6) of Torrence and Webster (1998)]The ACF of the AR1 model can then be calculated as [(A9) of Torrence and Webster (1998)]In the case of seasonal growth rate forcing only, the noise variance

### c. Method 3: Full solution to the Langevin equation

^{−1}), and

*A, D,*

### d. Method 4: Asymptotic solution to the Langevin equation

To understand the seasonal PB qualitatively, an ACF is derived from the asymptotic solution of the seasonal Langevin equation in the limit of weak seasonal forcing and weak damping (*B* ≪ 1, *BA* ≪ 1, *BD* ≪ 1) using the analytical solution of the ACF (2.18).

### e. Method 5: AR2 model simulation

*m*derived in (C.1). The residual noise is calculated with the observational time series asThe seasonal cycle of the noise variance

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^{1}

The effective amplitude *M* also depends on the phase difference between noise and growth rate forcings

^{2}

PB is the extreme persistence decline *τ* > 0, which, during certain seasons, is smaller than the absolute maximum of persistence decline at the boundary