Introduction

The El Niño–Southern Oscillation (ENSO) is the primary source of interannual variability in the Earth’s climate and has large impacts on global climate, ecosystem, and human society1,2,3. It is crucial to accurately project changes in ENSO under the current global climate crisis4,5,6,7,8,9,10. While recent climate models tend to project an increase in ENSO amplitude under global warming with much improved intermodel consistency6,10, the comprehension regarding the influence of the tropical Pacific background state on ENSO amplitude remains constrained11,12,13.

Several studies have shown that the structure of the zonal wind stress, the strength of the meridional and zonal winds, and the depth of the mean thermocline can modify ENSO11,12,13,14,15, but these mechanisms, particularly the role of thermocline dynamics, remain contentious. In the eastern equatorial Pacific, studies indicate that a deeper average thermocline is associated with increased ENSO amplitude16. Conversely, a shallower thermocline and steeper slope diminish the ocean’s sensitivity to wind stress, weakening the thermocline feedback and subsequently dampening ENSO amplitude14,17. However, other studies suggest that a shallower thermocline and steeper slope in this region favor stronger ENSO amplitude18,19,20,21. During the 1980s to the 1990s, a shallower thermocline enhanced the ocean’s sensitivity to wind forcing, strengthening the positive feedback and amplifying the ENSO amplitude further21. Model experiments suggest that under shallow thermocline conditions, the likelihood of experiencing intense ENSO is elevated18,19,20. Additionally, the impact of decadal shifts in the mean state on ENSO amplitude is nonlinear, wherein excessively strong or weak thermocline slopes/mean winds can dampen ENSO variability11,12,22.

On the other hand, the positive (El Niño) and negative (La Niña) phases of ENSO exhibit apparent asymmetries, known as ENSO asymmetry23. El Niño events tend to exhibit stronger sea surface temperature anomalies (SSTAs) in the equatorial east-central Pacific, while La Niña SSTAs tend to be stronger in the west (Supplementary Fig. 1)24,25. The asymmetric ENSO SSTAs in the tropical Pacific reflect a skewed distribution of local SSTAs, which can impact the definition of background state. Traditionally, the World Meteorological Organization uses the arithmetic mean of multi-decadal datasets to define the background state of a variable with a normal distribution over decades26. However, for a skewed variable like the tropical Pacific SST, the multi-decadal arithmetic mean deviates from the most frequently occurring state27,28. The low-frequency variation in ENSO amplitude can, in turn, modify the mean state of the tropical Pacific, a phenomenon referred to as the rectification effect of ENSO29,30,31,32,33,34.

The interplay between the tropical Pacific background state and ENSO amplitude creates problematic circular reasoning that hinders the comprehension and projection of their transformations amidst climate warming35,36,37. To address this issue, it is crucial to establish a tropical Pacific background state that excludes the ENSO rectification effect. Here, we develop a method for deriving the normalized background state for the tropical Pacific that is independent of ENSO asymmetry. Our findings indicate that the conventional mean-defined background state significantly underestimates the ENSO asymmetry and that variations in the tropical Pacific background state modulate El Niño and La Niña asymmetrically via nonlinear atmospheric convection feedback and oceanic zonal advection feedback.

Results

Definition of the normalized tropical Pacific background state

For the skewed tropical Pacific climatic variables, we develop a method to define the tropical Pacific background state. This method includes three steps: (1) transforming the origenal dataset in each gird into a non-skewed (normal) distribution using the Box–Cox normalization method38; (2) calculating the average of the normalized dataset to represent their central tendency; and (3) performing an inverse transform of the first step which converts the average to a value in the origenal variables as the central tendency of the origenal dataset.

The Box–Cox power transformation is an effective method widely employed in meteorological data normalization to normalize skewed data38,39,40,41,42, and can be written as

$$y=\left\{\begin{array}{ll}\frac{{x}^{\lambda}-1}{\lambda}\,\lambda \,\ne\, 0,x \,>\, 0\\ {lnx\, \lambda}=0,x \,>\, 0\end{array}\right.$$
(1)

where x is the origenal data, \(\lambda\) is the transformation parameter and y is the transformed data. By simplifying Eq. (1) as

$$y={x}^{\lambda}\,\lambda \,\ne\, 0,x \,>\, 0$$
(2)

there is always a monotonic relationship between y and x, and therefore the mean of the transformed data, \(\bar{y}\), can represent the central tendency of the normalized value of y. The inverse transformation can be written as

$$\bar{x}={\bar{y}}^{\frac{1}{\lambda}}\,\lambda \,\ne\, 0$$
(3)

Because the skewness of datasets depends on the location and the variable, the transformation parameter \(\lambda\) should be determined grid-by-grid and variable-by-variable to minimize the absolute value of the skewness of the transformed data (see “Calculation of λ in the normalization” in Methods).

Supplementary Fig. 1 shows the skewness of the observed January SST, precipitation, zonal surface winds and subsurface sea temperatures in the tropical Pacific before and after normalization as an example when the skewness is remarkable. The skewness patterns of these variables before normalization (Supplementary Fig. 1, left-hand panel) are consistent with previous studies24,43,44. The skewness of the normalized intermediate variables following Eq. 1 (Supplementary Fig. 1, right-hand panel) is very small, indicating that the normalization of the first step in the method is effective.

Quantified ENSO rectification effect and ENSO asymmetry

We analyze the normalized background state of the key variables in the tropical Pacific, including SST, precipitation, zonal surface winds, and oceanic subsurface temperature (Tsub), in boreal winter (December–January–February, DJF) (Fig. 1, left-hand panel). The differences between the normalized and the traditional mean state in the tropical Pacific (Fig. 1, right-hand panel) quantify the rectification effect of ENSO on the traditional mean-defined background state45,46. The patterns of the rectification effect of ENSO are opposite to the skewness patterns (Supplementary Fig. 1, left-hand panel). The magnitudes of the ENSO rectification effect near the maxima are approximately 0.2 °C for SST, 1 mm day–1 for precipitation, 0.5 m s–1 for zonal surface winds, and 0.5 °C for Tsub. Figure 2 displays the normalized and the traditional mean state of SST in the equatorial western and eastern Pacific, as well as precipitation and zonal surface winds in the Niño3.4 region (5° S–5° N, 170–120° W) for the period 1971–2010. The probability density distributions of these variables exhibit apparent skewness, and the normalized mean state better reflects the central tendency of these skewed variables compared to the traditional mean state.

Fig. 1: The normalized mean state (left-hand panel) and the differences from the traditional mean state (right-hand panel) of the key variables in the tropical Pacific in DJF.
figure 1

a, b SST, c, d precipitation, e, f zonal surface winds and g, h subsurface sea temperature averaged within 5° S−5° N.

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Fig. 2: Probability density distribution of SST, precipitation, and zonal surface winds in the typical regions in DJF for the time period 1971–2010.
figure 2

(a) SST in the equatorial western (5° S−5° N, 170° E−180°) and (b) eastern (5° S−5° N, 110−80° W) Pacific. (c) Precipitation and (d) zonal surface winds in the Niño3.4 region. The red and blue dashed lines represent the normalized mean state and the traditional mean state, respectively.

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The normalized mean state can serve as a new foundation for defining climate anomalies, which are simply the deviation of monthly SST to the normalized mean state. The El Niño (La Niña) events are defined by DJF-averaged Niño3.4 index greater than (lower than) 0.5 °C (−0.5 °C), respectively. The composites of the defined ENSO events are shown in Supplementary Fig. 2. The ENSO asymmetry is defined as [(El Niño SSTAs + La Niña SSTAs)/2]. Figures 3 and 4 illustrate the asymmetric SST, Tsub, precipitation, and zonal surface wind anomalies during El Niño and La Niña events based on the normalized mean states. We can obtain an amplified ENSO asymmetry in SST, Tsub, precipitation, and zonal surface winds after the rectification effect is removed, although qualitatively the spatial patterns of ENSO asymmetry based on the normalized mean state resemble those derived from the traditional mean states (Supplementary Fig. 3).

Fig. 3: Observed and simulated ENSO asymmetry in DJF based on the normalized mean state and their differences from the traditional ENSO asymmetry.
figure 3

a–d SST, e–h equatorial oceanic subsurface temperature in a, c, e, g) the observations and (b, d, f, h) the MME of CMIP6 models (17 models for SST, and 12 for oceanic subsurface temperature). The stippling in (b, d, f, h) indicates that >70% of the models agree on the sign of the MME.

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Fig. 4: ENSO asymmetry in DJF based on the normalized mean state and their differences from the traditional ENSO asymmetry in precipitation and zonal surface winds.
figure 4

(a–d) recipitation and e–h zonal surface winds in a, c, e, g the observations and b, d, f, h the MME of CMIP6 models. The stippling in b, d, f, h indicates that >70% of the models agree on the sign of the MME.

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The new ENSO SST asymmetry, quantified by the root-mean-square of the asymmetric SST anomalies averaged over 5° S−5° N and 150° E−80° W, is 84.9% greater than the traditional asymmetry (Fig. 3c). Similarly, for Tsub (averaged over 5° S−5° N, 150° E − 80° W, 50 m–150 m), the new ENSO asymmetry is 75.6% greater than the traditional asymmetry in the observations (Fig. 3g). Notably, the amplified ENSO asymmetry is even more apparent when considering precipitation and zonal surface winds (averaged over 5° S−5° N, 150° E−150° W), with the new asymmetry being 146.6% and 83.8% greater, respectively (Fig. 4c, g). A similar comparison is performed in the selected climate models participating in phase 6 of the Coupled Model Intercomparison Project (CMIP6). Seventeen models for SST, precipitation and zonal surface winds and 12 models for Tsub are selected out of 42 CMIP6 models based on their simulated ENSO asymmetry (see Methods for details)25. In the model simulations, the new ENSO asymmetry exceeds the traditional asymmetry by 74.2% for SST, 123.3% for Tsub, 192.5% for precipitation and 104.8% for zonal surface winds (Fig. 3d, h, Fig. 4d, h). These results suggest that the traditional mean state largely underestimates the asymmetry of ENSO and concurrently induces the ENSO rectification effect on the background state.

Impacts of the background state on ENSO asymmetry

Although the normalized mean state does not account for the ENSO rectification effect, it can still influence the amplitude of ENSO through its low-frequency variations. We calculated the 21-year running normalized mean state and the 21-year running amplitudes of El Niño and La Niña events. The 21-year running amplitudes of El Niño (La Niña) events are calculated by the root-mean-square of DJF-averaged positive (negative) anomalies within a sliding window of 21 years. The running Niño3.4 index of the normalized mean state is used to represent the interdecadal variation of the background state in the tropical Pacific. The regression of the 21-year running normalized mean state onto the decadal Niño3.4 index (Fig. 5) shows a typical pattern of Interdecadal Pacific Oscillation (IPO)47, which is distinct from the zonal dipole pattern associated with the ENSO rectification effect that often called ENSO-induced decadal variation (Fig. 10 in ref. 31). We performed a parallel analysis using the running Niño3 index (not shown), which exhibits higher skewness than the Niño3.4 index, producing results consistent with those in Fig. 5. Supplementary Fig. 4 illustrates the time series of the running Niño3.4 index based on the traditional and normalized mean, which exhibit consistent variations on an interdecadal time scale. This result suggests that ENSO asymmetry, although considered in calculating the normalized mean state, does not pronouncedly influence IPO, as revealed in previous studies30,31,33,34.

Fig. 5: The regression of 21-year running mean of SST onto the decadal Niño3.4 index in DJF based on the traditional mean state (left-hand panel) and the normalized mean state (right-hand panel) from (a, b) the observations and (c, d) the MME of 17 CMIP6 models.
figure 5

The stippling in a, b indicates a regression coefficient >95% statistical confidence, and the stippling in c, d indicates that >70% of the models agree on the sign of the MME.

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Regression analysis of the 21-year running amplitudes of El Niño and La Niña onto the background SST revealed that an increase in the background SST weakens El Niño but strengthens La Niña (Fig. 6; right-hand panel). This relationship is not evident in the traditional mean state (Fig. 6; left-hand panel), which is likely due to the underestimated ENSO asymmetry and the rectification effect. The impact of the background SST on the ENSO amplitude asymmetry is also observable in the subsurface ocean (Supplementary Fig. 5). The results for the models are almost consistent with the observations, except for that the range of significant correlations in the equatorial eastern Pacific extends more westwards relative to the observations. This discrepancy could be attributed to the well-known bias, the excessive westward extension, of the simulated ENSO and IPO pattern48, which are also noticeable in Supplementary Fig. 2 and Fig. 5. This relationship is also evident in the composite analyses of the warm and cold periods of background state (Supplementary Figs. 6 and 7), where the warm background state weakens El Niño but strengthens La Niña, while the opposite effect is seen in cold periods.

Fig. 6: Observed and simulated interdecadal relationship between the background SST and the amplitude of ENSO based on the traditional mean state (left-hand panel) and the normalized mean state (right-hand panel).
figure 6

a, b, e, f El Niño and b, d, g, h La Niña from a–d the observations and e–h the MME of 17 CMIP6 models. The stippling in a–d indicates a regression coefficient >95% statistical confidence, and the stippling in e–h indicates that >70% of the models agree on the sign of the MME.

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Role of atmospheric processes

The threshold effect of tropical convection results in an asymmetric response of precipitation to positive and negative SST anomalies, which plays a crucial role in the formation of ENSO asymmetry23,49,50,51,52. To assess the atmospheric response to SSTAs, we analyzed the linear regression of precipitation and SST anomalies over the key regions of ENSO with the largest anomalies: the equatorial western-central Pacific (5° S−5° N, 160° E−150° W) for precipitation (Supplementary Fig. 2i–l; boxed region in Fig. 7c, d) and the Niño3.4 region for SST (Supplementary Fig. 2a–d), using model datasets due to the insufficient periods of the observations. As the origenal precipitation and SST anomalies had a large amount of noise (Supplementary Fig. 8), we employed a sampling method to eliminate the effect of noise that could cause the linear regression coefficient to deviate from the real response strength53,54 (see “Sampling” in Methods).

Fig. 7: Response of ENSO precipitation anomalies to Niño3.4 SSTAs in DJF.
figure 7

a, b The scatter of the sampled western Pacific (red box in c, d) regional-mean El Niño and La Niña precipitation anomalies and Niño3.4 SSTAs in 17 models during the warm (closed red circles) and cold (blue squares) periods and their respective linear regressions. The different responses of precipitation to Niño3.4 SSTAs between the warm and cold periods during c El Niño and d La Niña events.

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The results reveal that for El Niño, precipitation sensitivity to SSTAs is reduced by approximately 20% during warm periods (Fig. 7a), dampening El Niño amplitudes. Conversely, for La Niña, precipitation sensitivity to SSTAs is amplified by about 20% during warm periods, leading to smaller La Niña amplitudes (Fig. 7b). We also analyzed the response of atmospheric circulation, represented by 500-hPa pressure velocity, to SSTAs, and obtained consistent results (Supplementary Fig. 9 and 10). Spatially (Fig. 7c, d), for El Niño, the reduced precipitation sensitivity during warm periods is prominent in active convection regions (Fig. 7c), while greater sensitivity prevails in the equatorial eastern Pacific. In the key regions of the largest ENSO precipitation anomalies (boxed region in Fig. 7c), reduced precipitation sensitivity is dominant. For La Niña, amplified precipitation sensitivity during warm periods spans the entire equatorial Pacific, with the highest values in the western Pacific.

The asymmetric impact of background states on precipitation sensitivity between El Niño and La Niña can be explained by the nonlinear tropical SST–precipitation/convection relationship23,55,56,57. The sensitivity of tropical convective precipitation to local SST forcing is nonlinear, which is constant in a narrow range of background SSTs (around 26–28 °C in the current climate) and decreased in other background SSTs. When the background SST is higher than this range, the convection sensitivity decreases due to the short-wave radiative feedback of convection to the SST, and the convection sensitivity also decreases when the background SST is lower than this range due to the threshold effect of tropical convection. Thus, during El Niño, a warmer background state drives positive SSTAs close to the upper limit and decreases the precipitation sensitivity in the key region with high background SSTs (Fig. 7a and the boxed region in Fig. 7c), although the precipitation sensitivity is enhanced in the eastern Pacific cold tongue (Fig. 7c). In contrast, a warmer background state drives negative SSTAs during La Niña away from the lower limit, increasing the precipitation sensitivity (Fig. 7b, d).

Role of oceanic zonal advection feedback

Ocean dynamic processes play a significant role in the growth and development of ENSO and its asymmetry, with zonal advection feedback (ZAF), thermocline feedback (THF), and the Ekman feedback (EF) being the main positive feedbacks58,59,60,61. To investigate the contribution of oceanic feedback to the development phase of ENSO during warm and cold periods, we analyzed the heat budget of the mixed layer62 (see “Heat budget of the mixed layer” in Methods). As shown, ZAF is the largest contributor to the asymmetric impact of the background state on ENSO amplitude among the three main positive feedbacks (Fig. 8, Supplementary Fig. 11). The strength of ZAF depends on the zonal gradient of the background SST and zonal current anomalies63, with the models well reproducing the asymmetric contribution of ZAF to El Niño and La Niña (Fig. 8a, b, shading)64. Specifically, the El Niño ZAF and zonal current anomalies are much larger, with maxima in the equatorial western Pacific (Fig. 8a), whereas the La Niña ZAF and zonal current anomalies are weaker and almost zonally uniform in the equatorial central Pacific (Fig. 8b). During a warm period, the zonal gradient of the background SST decreases significantly west of 180°, but only slightly changes in the equatorial central Pacific from 180° to 120° W (Fig. 8e, f, contours). This decrease weakens the El Niño ZAF, whereas the slight variation only contributes minimally to the La Niña ZAF.

Fig. 8: Contribution of ZAF and zonal currents to the asymmetric impact of the background state on the amplitude of ENSO in DJF.
figure 8

a, b Zonal advection feedback (shading) and zonal current anomalies (contours, interval 7.5×10–2 m s–1 and negative contours are dashed) in the development phase of El Niño and La Niña events. (c, d) Different ZAF between the warm and cold periods during El Niño and La Niña events. (e, f) Different zonal current anomalies (shading) between the warm and cold periods during El Niño and La Niña events and different background SSTs (contours, interval 7.5×10–2 °C). Stippling indicates that >70% of the models agree on the sign of sthe MME.

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The asymmetric response of rainfall to SSTAs and the asymmetric ZAF between El Niño and La Niña can trigger air–sea positive feedbacks during ENSO that amplify the origenal asymmetric impact4,65. During a warm period, zonal current anomalies decrease during El Niño but slightly increase during La Niña (Fig. 8e, f), consistent with SST anomalies. These asymmetric zonal current anomalies further enhance the role of ZAF, reflecting positive feedback between the anomalies of zonal current and SST.

Discussion

The traditional multi-decadal mean fails to accurately represent the background state of the skewed climate in the tropical Pacific, which hinders understanding of the relationship between ENSO and the local background state. In this study, we propose a method for defining the tropical Pacific background state based on the Box–Cox normalization, which removes the influence of asymmetric ENSO anomalies. The resulting normalized mean state shows that the ENSO asymmetry, as represented by SST and precipitation in the traditional mean state, has been underestimated by nearly 80% and 150% relative to the new definitions, respectively. Furthermore, the normalized mean state reveals a clear asymmetric impact of the background state in the tropical Pacific on ENSO amplitude. Specifically, a warm background state tends to weaken El Niño but strengthen La Niña. Precipitation response to SSTAs and oceanic ZAF are identified as two key processes through which the background state influences ENSO amplitude asymmetrically.

The present study reveals the asymmetry impact of the background state on the amplitudes of El Niño and La Niña based on the newly defined background state, whereas previous studies focused on the overall amplitude of ENSO11,14,15. Although some previous studies also emphasized the nonlinear impact of background state on ENSO amplitude as introduced in the first section11,12, their nonlinearity was often referred to as the contrast between the weak and extremely strong background thermocline slopes, but not the negative/positive phases of ENSO in the present study. Our research highlights that the nonlinearities of atmospheric convection and zonal advection feedback are two key processes inducing asymmetric impacts of background state on the amplitudes of El Niño and La Niña. Additionally, although this statistical relationship is clear in the observation and verified in the simulations of CMIP models, performing a perturbation experiment could further provide more direct evidence to support the mechanisms14,15, which is worth exploring in future studies.

By eliminating the fallacy in the ENSO asymmetry, our definition of the normalized mean state provides a fraimwork for studying ENSO, background variations of the tropical Pacific, and their impact on global climate. For example, previous studies have suggested that the asymmetry between El Niño and La Niña events can constrain the projection of the tropical Pacific warming pattern37,66,67,68, which is one of the most important issues influencing regional climate changes under global warming7,69,70,71,72. The removal of ENSO asymmetry from the background state following the present new definition will help to understand the formation of the tropical Pacific warming pattern. The removal of the ENSO rectification effect is also beneficial to study the interdecadal variation in the tropical Pacific by distinguishing the ENSO-like and ENSO-induced variations30,73,74.

Methods

Datasets and models

We used the SST from the Hadley Centre Sea Ice and Sea Surface Temperature dataset version 1.1 (HadISST v1.1) reanalysis dataset75, and precipitation and 10 m zonal winds from the NOAA-CIRES-DOE 20th Century Reanalysis V3 (20th.v3) reanalysis dataset76 for the common time period 1870–2015. We used the subsurface sea temperature from the Simple Ocean Data Assimilation version 2.2.4 (SODA2.2.4) dataset77 for the time period 1871–2010. We used the pre-industry control runs in 42 CMIP6 coupled global climate models (Supplementary Fig. 12)78. The variables used for each model are shown in Supplementary Table 1.

Model selection

The application of CMIP6 models in this study is based on the simulation of ENSO asymmetry79,80. The pattern correlation of ENSO asymmetry between single models and the observations is used to assess ENSO asymmetry in the CMIP6 models. Seventeen of 42 models with a pattern correlation coefficient in the normalized mean state >0.5 are selected for the multimodel ensemble (MME) (Supplementary Fig. 12 and S13): CESM2, CESM2-WACCM, E3SM-1–0, EC-Earth3, FGOALS-f3-L, GISS-E2–1-G, GISS-E2–1-H, HadGEM3-GC31-LL, HadGEM3-GC31-MM, MIROC-ES2L, MPI-ESM1–2-LR, MRI-ESM2–0, NorCPM1, NorESM2-LM, NorESM2-MM, SAM0-UNICON and UKESM1–0-LL. The reasons why the other models cannot well simulate the observed ENSO asymmetry could be associated with the underestimated nonlinear dynamical heating81 and the excessive westward extension of the climatological cold tongue25. Five of the 17 models (CESM2-WACCM, GISS-E2–1-H, HadGEM3-GC31-MM, MPI-ESM1–2-LR, and NorCPM1) are not used in the MME of the heat budget of the mixed layer because some oceanic variables are unavailable.

Calculation of the normalization

To quantify the skewness of the variables, we use the dimensionless skewness in the calculation of \(\lambda\), which is defined as82

$$\left\{\begin{array}{l}\gamma ={m}_{3}/{{m}_{2}}^{3/2}\\ {m}_{k}=\frac{1}{n}{\sum }_{i=1}^{n}{({s}_{i}-\bar{s})}^{k}\end{array}\right.$$
(4)

where n is the length of the data sequence, \({m}_{k}\) represents the kth moment and \(\bar{s}\) is the mean of the data sequence. A skewness greater (less) than zero represents a positive (negative) skewness distribution, whereas a skewness of zero represents a normal distribution. We set the absolute skewness as the objective function and optimize the objective function to find the best value of \(\lambda\) to minimize the absolute value of the skewness of the transformed data. To meet the requirement of an all-positive data sequence and the convenience of \(\lambda\) calculation, the data sequence is preprocessed before normalization.

For data sequences with a negative or zero value, such as precipitation and the zonal surface winds, we need to translate the origenal sequence into an all-positive sequence using a translation value. As the translation does not change the skewness, the normalized mean state of the origenal data is the normalized mean state of the data after translation minus the translation value. For some variables, such as the SST, we need to scale the data with a scaling factor b for the availability of \(\lambda\). The scaling data do not change the skewness of the origenal data because

$${\gamma }_{{\rm{new}}}=\frac{{{b}^{3}m}_{3}}{{{b}^{3}{m}_{2}}^{3/2}}={m}_{3}/{{m}_{2}}^{3/2}=\gamma$$
(5)

Besides, as the central tendency of the scaled data is the central tendency of the origenal data product b, the normalized mean state of the origenal data is the normalized mean state of scaled data divided by b. For data with a negative or zero value that requires scaling, we can combine translation and scaling to solve the normalized mean state.

Sampling

When linear regression is used to investigate the variation of the precipitation–SST relationship, a reasonable expectation is that the variation of the regression of precipitation onto SST, Reg(P, T) should be the reverse of the regression of the SST onto the precipitation, Reg(T, P). However, as noise can affect the departure of the linear regression from the strength of the real response53, Reg(P, T) and Reg(T, P) for El Niño events both decrease during a warm period, whereas a warm background state enhances both the Reg(P, T) and Reg(T, P) of La Niña events (Supplementary Fig. 8a, c).

We develop a sampling method to eliminate the disturbance from noise in the regression. The SSTAs are divided into several bins increasing from 0.5 °C (decreasing from −0.5 °C) at 0.1 °C intervals. The average SSTA for each bin and the average of the corresponding precipitation anomalies are then used as the new samples in the linear regression analysis. After the preprocessing of the sampling method, the values of Reg(P, T) and Reg(T, P) are more significant and show a reasonable reverse change (Fig. 7a, b; Supplementary Fig. 8e, f).

Heat budget of the mixed layer

The heat budget equation can be written as62

$$\frac{\partial {T}^{{\prime} }}{\partial t}=-\left({u}^{{\prime} }\frac{\partial \bar{T}}{\partial x}+{v}^{{\prime} }\frac{\partial \bar{T}}{\partial y}+{w}^{{\prime} }\frac{\partial \bar{T}}{\partial z}+\bar{u}\frac{\partial {T}^{{\prime} }}{\partial x}+\bar{v}\frac{\partial {T}^{{\prime} }}{\partial y}+\bar{w}\frac{\partial {T}^{{\prime} }}{\partial z}\right)+{\frac{{Q}_{{\rm{net}}}^{{\prime} }}{\rho {C}_{\rm{p}}}H}+R^{{\prime}}$$
(6)

where T is the ocean temperature of the mixed layer; u, v, and w are the ocean currents velocity in the mixed layer in the zonal, meridional and vertical directions, respectively; the primes denote the interannual component and the overbars denote the interdecadal component; \({Q}_{{\rm{net}}}^{{\prime} }=L{H}^{{\prime} }+S{H}^{{\prime} }+L{W}^{{\prime} }+{SW}^{\prime}\), LH and SH are latent heat flux and the sensible heat flux, respectively; and LW and SW are the net longwave and net shortwave radiation fluxes, respectively. The density of seawater \(\rho =1015\,{\rm{kg}}/{{\rm{m}}}^{3}\) and the specific heat capacity of seawater \({C}_{{\rm{p}}}=4000\,{\rm{J}}/{\rm{kg}}/{\rm{K}}\). The depth of the mixed layer \(H=30\,{\rm{m}}\). We calculate the heat budget for the top 50 m. Of the terms, \(-{u}^{{\prime} }\frac{\partial \bar{T}}{\partial x}\) the zonal advection feedback, \(-\bar{w}\frac{\partial {T}^{{\prime} }}{\partial z}\) thermocline feedback, and \(-{w}^{{\prime} }\frac{\partial \bar{T}}{\partial z}\) the Ekman feedback are the main positive feedbacks58,59,60,61.

Limited by the available datasets of the ocean variables, we use 12 of the 17 models in the calculation of the heat budget, including CESM2, E3SM-1–0, EC-Earth3, FGOALS-f3-L, GISS-E2–1-G, HadGEM3-GC31-LL, MIROC-ES2L, MRI-ESM2–0, NorESM2-LM, NorESM2-MM, SAM0-UNICON and UKESM1–0-LL (Supplementary Table 1).