To what extent does ENSO rectify the tropical Pacific mean state?

The role of the El Niño-Southern Oscillation (ENSO) in modulating the mean sea surface temperature (SST) in the tropical eastern Pacific is investigated. A strategy is developed to separate the observational record from 1958 to 2010 into two groups, ENSO and non-ENSO periods. A simple analytical framework is constructed to quantitatively delineate the contributions of oceanic dynamic heating (ODH) and surface heat fluxes to rectifying the mean SST. It is found that the differences in the mean SST between the two periods are evident, though minor, despite distinctive interannual SST variabilities. Both linear and nonlinear ODH, as well as surface heat fluxes, contribute to this slight mean SST difference. Idealized oceanic model experiments in the presence and absence of ENSO are conducted. The experiments confirm that ENSO moderately impacts the mean SST in the equatorial eastern Pacific. Although the amplitude of the linear ODH associated with ENSO is large, its impact on the long-term mean SST is small because the sign of its contribution changes between El Niño and La Niña phases. The nonlinear ODH, on the other hand, has the same sign during the warm and cold episodes. However, its accumulated effect on the mean SST is small due to its weak amplitude.


Introduction
The El Niño-Southern Oscillation (ENSO) is the most energetic interannual variation of the climate system on Earth, with significant environmental and societal impacts worldwide (Philander 1990). It is characterized by strong sea surface temperature (SST) and surface wind fluctuations in the tropical Pacific (Rasmusson and Carpenter 1982). While the growth of the ENSO involves positive ocean-atmosphere feedbacks (Bjerknes 1969;Philander et al. 1984;Hirst 1986Hirst , 1988Li 1997a), its oscillation results from a delayed oscillator involving equatorial ocean waves (e.g., Suarez and Schopf 1988;Battisti and Hirst 1989) and a recharge oscillator involving the zonal mean thermocline variation (Jin 1997;Li 1997b). The tropical climate state is a primary factor in determining an atmosphere-ocean coupled stability for the ENSO system (Li 1997b;An and Jin 2000;Fedorov and Philander 2000;An et al. 2020).
It has been shown that the Pacific mean state, with a pronounced cold tongue (warm pool) to the east (west), a zonally tilting thermocline structure with a strong vertical mean temperature gradient to the east, is critical in determining the preferred location of maximum SST variability associated with the ENSO (Philander 1990;Li and Hsu 2018). While the annual mean solar radiation is zonally symmetric, the east-west asymmetry of the Pacific mean climate arises from the forcing of trade winds caused by the rotation of the Earth and is amplified by various positive ocean-atmosphere feedbacks Li 1997a). The east-west asymmetry in the mean SST field is accompanied by pronounced easterlies at the equator, as part of the mean Walker Circulation, and strong mean upwelling and zonal and vertical temperature gradients in the eastern and central equatorial Pacific, all of which permit an effective air-sea coupling on the interannual timescale in the eastern equatorial Pacific, leading to the ENSO growth. The positive feedbacks relevant here include zonal advective feedback and Ekmanpumping-induced upwelling feedback (Li 1997b).
The impact of the mean state on ENSO can also be inferred from its phase locking to the annual cycle. It has been noted that the mature stage of ENSO often occurs in northern winter (DJF). This implies that ENSO may be modulated by the climatological annual cycle, as SST in the eastern equatorial Pacific exhibits a remarkable annual cycle, with its amplitude comparable to ENSO . Chang et al. (1994) showed that the ENSO phase locking might result from its nonlinear interaction with the annual frequency. On the other hand, Li and Hsu (2018) demonstrated that even in a linear model with a seasonally varying mean state, the ENSO phase locking resembling the observed can still occur. In such a model, the sole cause is a so-called season-dependent coupled instability (Li 1997b).
There are some previous studies suggest that many aspects of the ENSO vary on decadal-interdecadal time scales in association with changes in tropical Pacific mean climate (e.g., Wang 1995;An and Wang 2000;Philander 2000, 2001;Wang and An 2001;An and Jin 2001;Choi et al. 2012;Chung and Li 2013;Guan and McPhaden 2016;Ogata et al. 2013;Sun and Okumura 2020;Power et al. 2021, among others), yet only a few studies have been devoted to investigating the ENSO feedback to the mean climate.
As a dominant interannual variability in the tropical Pacific, does the ENSO modulate the Pacific mean state? There were some empirically based studies to address this problem. For instance, a residual due to El Niño and La Niña amplitude asymmetry during their mature phase was suggested to play a role in modulating the mean state (Rodgers et al. 2004;Sun and Yu 2009;Choi et al. 2012;Yu and Kim 2011;Ogata et al. 2013), but such a hypothesis is faulted as the asymmetry at one-time level cannot determine the outcome of the long-term average result. Note that observed SST anomalies (SSTAs) in the equatorial eastern Pacific (EEP) exhibit a marked asymmetric evolution between El Niño and La Niña (Chen et al. 2016). While El Niño is characterized by a rapid decay after its peak and a fast phase transition to a cold episode in the following winter, La Niña often persists to the second and even third year (Ohba and Ueda 2007;Okumura and Deser 2010;Choi et al. 2013;DiNezio and Deser 2014;An and Kim 2018).
A possible process through which the ENSO modulates the mean state is through a nonlinear rectification effect (Sun and Zhang 2006;Liang et al. 2012;Sun et al. 2014;Hua et al. 2015). Kessler and Kleeman (2000) proposed the concept of nonlinear rectification in examining the nonlinear ocean dynamical impact on ENSO by MJO-related high-frequency wind fluctuations. With the aid of a simple low-order nonlinear model, Liang et al. (2012) found that the Pacific mean state, particularly cold tongue mean SST, could be significantly modulated by enhanced ENSO variability under global warming. They indicated that the rectification arose from nonlinear advective processes. While the result from the simple box model is interesting, it is worth mentioning that the procedure regulating the SST in the model is too simple. Two boxes described the equatorial western and eastern Pacific SST change. The model's only nonlinear advective processes regulating the eastern (western) Pacific SSTA are vertical (zonal) advection. Therefore, it is necessary to develop a new strategy to reveal the ENSO nonlinear rectification effect on the mean state from an observational analysis perspective and an idealized coupled general circulation modeling perspective.
Motivated by the abovementioned consideration, we intend to develop a methodology to clarify the ENSO nonlinear rectification effect on the mean state in this study. The remaining part of this paper is organized as follows. Section 2 describes the observational data used in this study, the observational analysis strategy, and coupled model experiment designs. The main results concerning the differences between the period mean states with and without ENSO variability in the real-world observation are presented in Sect. 3. The results from numerical experiments are illustrated in Sect. 4. A summary is provided in Sect. 5.

Data and method
Observational data used for the current study include Ocean Reanalysis System 5 (ORAS5) (Zuo et al. 2019), the fifthgeneration atmospheric reanalysis ERA5 (Hersbach et al. 2018), the Simple Ocean Data Assimilation (SODA) (Carton and Giese 2008), version 2.2.4 products (SODAv2.2.4), and Japanese 55-year Atmospheric Reanalysis (JRA-55) products (Kobayashi et al. 2015). These datasets contain the three-dimensional ocean current and temperature fields (the vertical currents are calculated by the oceanic continuity equation), surface atmospheric wind, temperature, geopotential height, specific humidity fields, and surface radiation and heat flux fields, with the period from the year 1958 to 2010. The ORAS5, which is a product of the global eddypermitting ocean-sea ice ensemble (5 members) reanalysisanalysis system, has an interpolated horizontal resolution of 1° by 1° in the tropics and 75 vertical levels with a 10-m resolution in the upper ocean, while the ERA5 has a regular global lat-lon grid of 0.25 degrees for the reanalysis and hourly temporal resolution. The SODA dataset has a 0.4° by 0.25° horizontal resolution and 40 vertical levels with 10-m spacing in the upper ocean. The JRA-55 has a horizontal resolution of 0.5° by 0.5° and a 6 h time interval. As for SST datasets, the Extended Reconstructed Sea Surface Temperature version 5 (ERSST V5) (Huang et al. 2017) from the US National Climate Data Center (NCDC), with a horizontal resolution of 2° by 2°, and HadISST (Rayner et al. 2003) from the Met Office Hadley Centre Sea Ice and SST dataset, with a horizontal resolution of 1° by 1°, are used, covering the period from 1958 to 2019. In addition, surface heat flux products such as surface net shortwave radiation, surface downward longwave radiation, and surface latent and sensible heat fluxes that are calculated with near-surface specific humidity and temperature and sea level pressure are from NOAA-CIRES 20th Century Reanalysis (20CRv2) (Whitaker et al. 2004;Compo et al. 2011), which are used for comparison with the ERA5 and the JRA-55. This dataset provides the first estimates of global atmospheric fields from 1871 to the present at a 6-hourly temporal interval and 2° spatial resolution. Intercomparisons with independent radiosonde data indicate that the reanalysis is generally of high quality (Compo et al. 2011).
Idealized numerical model experiments are designed to unveil the impact of the ENSO on the mean climate by contrasting the time-mean state of forced ocean general circulation model (GCM) experiments. The model used for these experiments is the Community Earth System Model 2.1 (CESM2.1) ocean component, Parallel Ocean Program version 2 (POP2). POP2 is a state-of-the-art general ocean circulation model that solves the 3-dimension primitive equations of rotational fluid dynamics and thermodynamics with standard approximations of Boussinesq and hydrostatics. It is developed and maintained at Los Alamos National Laboratory (Smith et al. 2010) and is the ocean component of the fourth version of the Community Climate System Model (CCSM4) developed at the National Center for Atmospheric Research (Gent et al. 2011;Danabasoglu et al. 2012). The model uses the displaced-pole coordinate grid with the pole centered over Greenland and has a nominal horizontal resolution of 1°. The meridional spacing near the Equator is enhanced (to about 1/2°) to more accurately resolve equatorial waves and currents, including equatorial Kelvin/Rossby waves. Vertically it resolves 60 depth levels with resolution varying from 10-m in the upper ocean to 250 m in the deeper ocean (Lucas et al. 2013). For a detailed description of the model, readers are referred to Smith et al. (2010) and Danabasoglu et al. (2012).
In the control run (CTRL), the experiment is forced by JRA-55 daily atmospheric surface fields containing 10-m zonal and meridional wind, specific humidity, air temperature at 10-m, sea level pressure, precipitation, surface downward longwave radiation, and downward solar radiation. The subsequent sections show that the CTRL successfully simulates Niño3.4-SST evolution from 1958 to 2010. While in a sensitivity experiment (EXP_NO_IA), the model is forced by the JRA-55 daily surface fields for the same period with the interannual signals excluded by Butterworth 1 to 8 year band-pass filter. This sensitivity experiment aims to remove the ENSO signal so that the time mean states of the two runs can be compared. Another two experiments are designed to make a further comparison. In one of them, the long-term mean atmospheric forcing of 1958-2010 (EXP_AM) is applied, while in the other, the atmospheric forcing used is the long-term mean atmospheric forcing of 1958-2010 plus the interannual signals over the period (EXP_AM + IA). Thus, the time-mean temperature differences between the run with ENSO and the run without ENSO can delineate the rectification effect of ENSO events on the climate mean state.
The role of ENSO on the mean state, especially the impact of ENSO-induced nonlinear ocean heating, may be assessed directly through an observational analysis.
An ENSO year is defined from May 1 (or July 1) of its developing year to April 30 (June 30) of its decaying year. Assume that both the ENSO year (31 years) group and the normal (non-ENSO) year (21 years) group are approximately in an equilibrium state (i.e., their respective longterm SST tendencies vanish), one may derive the following balance between the ocean dynamics effect ( D ocn ) and the net surface heat flux effect ( Q net ): where denotes the time-mean difference between the ENSO state and the non-ENSO state, the first term on the left-hand side denotes the difference of the net surface heat flux that consists of five components as follows: where Q nsw denotes net shortwave radiation, Q ulw and Q dlw are upward and downward longwave radiation, Q lh is surface latent heat flux, and Q sh is surface sensible heat flux.
The difference in the upward longwave radiation at the ocean surface can be approximately written as where an overbar denotes the climatological mean state during 1958-2010 and T s denotes the sea surface temperature.
Surface latent and sensible heat fluxes may be calculated using the following bulk formulas, following Xie et al. (2010) and Zhang and Li (2014): where a is surface air density, L v is the latent heat of evaporation and C p is the specific hear capacity at constant pressure, C e and C h are the heat exchange coefficients of Q lh and Q sh respectively, V is surface wind speed, RH is surface relative humidity, and ΔT denotes the difference between sea surface temperature T s and near-surface air temperature T a . The Clausius-Clapeyron equation has been applied to derive Eq. (4), ln(q as ∕q s where q as is saturated specific humidity of surface air, = L v ∕RT s T a , R v is the ideal gas constant for water vapor.
The surface latent heat flux difference term may be further decomposed to two terms, with the first term being relevant to SST change: and the second term being a residual, reflecting atmospheric processes Similarly, the sensible heat flux difference term can be separated to the following two terms: and By substituting each of the flux terms into Eq. (1), the difference of long-term mean SST between the two states (i.e., ENSO and non-ENSO states) can be expressed as where the denominator A = 4 T s 3 + 1 Q lh + 2 V is the function of the climatological mean state. By diagnosing the relative contributions of five terms in the numerator based on the observations, one may determine to what extent the mean SST difference between the ENSO state and the normal state is determined by the difference of various heat flux terms (i.e., Q nsw , Q dlw , Q lha and Q sha ) and oceanic dynamics D ocn , latter of which consists of linear and nonlinear temperature advective processes. Through the diagnosis of Eq. (10), one may quantitatively estimate the effect of nonlinear temperature advection associated with the ENSO on rectifying the mean state.
The oceanic temperature advection term D ocn in Eq. (10) may be expressed as: where = 1 × 10 3 kg∕m 3 is the density of water, C w = 4 × 10 3 J∕kg∕K is the specific heat of water, h = 30m is used to approximate the climatological mixed-layer depth in the eastern equatorial Pacific, V = (u, v, w) represents the 3D ocean current, ∇ = ( ∕ x, ∕ y, ∕ z) denotes the 3D gradient operator.
To highlight the rectification effect of interannual variability, here we decompose the velocity and temperature fields into three components, V = V c + V i + V r , T = T c + T i + T r , where subscript c stands for the climatological mean annual cycle component, subscript i denotes the interannual component, and subscript r denotes the residual component. Hereafter D ii = V i ⋅ ∇T i stands for nonlinear dynamic heating, D ic = V i ⋅ ∇T c + V c ⋅ ∇T i represents linear dynamic heating, D cc = V c ⋅ ∇T c represents the climatological mean temperature advections, and represents the sum of the residual component related advective terms. Therefore, the sum of all terms above represents the total oceanic dynamic heating (ODH) as D tt = D ii + D ic + D cc + D rs . The Butterworth bandpass with a period of 1-to-8-year filter was applied to extract the interannual component from the original time series at each grid point.
It is worth mentioning that the quasi-equilibrium state approximation for the ENSO and non-ENSO states is validated based on the observational analysis shown in Table 2. The period-mean SST tendencies during the 31 ENSO and 21 normal years are close to zero, respectively.

ENSO impact on the mean state: observational diagnosis
This section intends to reveal to what extent the ENSO may impact the mean state from an observational diagnosis perspective. The simple equilibrium state model is used to diagnose the role of each of the dynamic and thermodynamic terms in Eq. (10) Fig. 1d, f, which are slightly different from the event years classified by ERSSTv5, but their defined event years overall are close to those defined by NOAA (Table 1). To clearly show the difference in the SSTA variability between the two equilibrium states, Fig. 1g, h illustrate the maps of the SSTA standard deviation during each of the two periods. The interannual SSTA variability is much greater (at least three times larger) in the ENSO years than in the NOML years.
Despite the distinctive interannual variabilities between the ENSO and NOML periods, the period-mean SST in the tropical Pacific is quite similar. Table 2 lists the periodmean SST averaged over the EEP during the entire 52-yr period, 21 NOML years, and 31 ENSO years, respectively. Whereas the mean SST during El Niño and La Niña years exhibit a 2 °C difference, the mean SST values during the ENSO and NOML years are almost identical, implying that the ENSO rectification effect on the mean state is very weak. The details of the period-mean SST differences between ENSO and NOML years are further shown in Fig. 2. Although there is a positive SST difference over the concerned EEP region, the amplitude is small, ranging from 0.16 to 0.52 according to the event years classified by ERSSTv5, Hadley SST, and NOAA (with ensemble SST) span 1958 to 2018 in both situations where one year is defined from May (0) to April (+1) and from July (0) to June (+1).
By diagnosing Eq. (10) with observational data, one may reveal the relative role of the ocean dynamics and surface heat fluxes in contributing to the mean SST difference. Ocean data from ORAS5, SODA, and atmosphere data from ERA5 and 20CRv2 from 1958 to 2010 are compared. Figure 3 shows the longitudinal distribution along the equator of the mean SST difference between the ENSO and NOML years and contributions from individual five terms displayed on the right-hand side of Eq. (10). The solid black curves in Fig. 3a-c, g-i show the observed mean SST difference between the ENSO and NOML years, whereas the black dash curves represent the mean SST difference diagnosed by different terms on the right-hand side of Eq. (10). By comparing the observed and estimated, one can conclude that the simple analytic model can reproduce the observed mean SST difference well in general. Besides, the SST difference values of each term averaged over the EEP box ( 4 o S ∼ 4 o N, 180 o ∼ 90 o W ) are displayed as Fig. 3d-f, j-l. These values are generally small (less than 0.25), primarily when the ENSO state is defined by ERSSTv5, which indicates that all dynamic and thermodynamic processes contribute equally to the mean SST difference. Within the dynamic and thermodynamic terms, the relatively more significant contributing factors are surface net shortwave radiation and surface downward longwave radiation, and they tend to cancel out each other. The oceanic dynamic heating (i.e., 3D temperature advection) is weak. To further understand the opposite effect of surface net shortwave and longwave radiative heating from the perspective of thermodynamics, we compared the spatial patterns of SST, net surface shortwave radiation, downward longwave radiation, and total cloud cover anomalies associated with El Niño and La Niña composites in Fig. 4a-c, e-g. Their linear relationships with the cloud cover and SST fields explain the horizontal patterns of the radiative fluxes. For El Niño, the warm SSTA in the equatorial Pacific (Fig. 4a) is accompanied by more cloud cover, which, on the one hand, reduces downward shortwave radiation and, on the other hand, increases downward longwave radiation (Fig. 4b-c). A mirror image appears for the La Niña composite. While for the ocean dynamic heating (ODH), positive (negative) heating is expected over the equatorial Pacific for El Niño (La Niña) composite with the large center located at the east (Fig. 4d, h). The overall impact of the combined El Niño and La Niña on  (0) to April (+ 1) (July (0) to June (+ 1)). b Is the same as (a), except the data is from Hadley SST. c-d Are yearly averaged SSTA (°C, from May (0) to April (+ 1)) in the eastern equatorial Pacific (EEP, 4S° ~ 4°N, 180° ~ 90°W) with SST data from ERSSTv5 and Hadley SST, respectively. Red bars are for El Niño years, while blue and gray bars are for La Niña and normal years. e-f Are the same as (c-d), except the yearly averaged SSTA is from July (0) to June (+ 1). g-h Horizontal distributions of the standard deviation of SSTA during the ENSO and NOML years. A black box denotes the EEP region Fig. 2 a The difference in period-mean SST (unit: °C) between the ENSO and NOML years along the equatorial region (averaged 4°S ~ 4°N) from 120°E to 88°W span 1958 to 2010, with one year defined from May (0) to April (+ 1) and blue, brown, and red lines denote the ENSO and NOML years that are classified by ERSSTv5, Hadley SST, and NOAA (ensemble SST) respectively. b is the same as a, except that one year is defined from July (0) to June (+ 1). c-d are the same as (a-b) but with a period from 1958 to 2018. e-j are the period-mean SST spatial differences span 1958 to 2010. A black box denotes the EEP region the mean state is shown in the third row of Fig. 4. Their net effect is a weak positive SST difference in the EEP (Fig. 4i). It is interesting because the number of El Niño years (15) is less than that of La Niña years (16). Nevertheless, the accumulated effect of El Niño years on the mean SST is slightly more significant than the La Niña accumulated effect. Consequently, the accumulated net impact of the shortwave and longwave radiations and the ODH fields resemble the El Niño composite, with much weaker amplitude (Fig. 4j-l). Interestingly, the periodmean SST, shortwave, longwave radiative flux, and ODH fields during the NOML years, with the climatological mean values removed, are small and have approximately the same patterns as those of the ENSO years but with the opposite sign (Fig. 4m-p).
Despite distinctive SSTA variabilities during the ENSO and non-ENSO years, the effect of the ODH, particularly nonlinear ocean dynamic heating, on the mean SST remains essentially the same. Figure 5a shows the time series of the total ocean temperature advection (or ODH) term and its decomposed nonlinear ODH term D ii , linear ODH term D ic , climatological ODH term D cc and residual term D rs averaged over the EEP. Note that the total ODH oscillates around the climatological annual cycle component D cc . Their long-term average has a negative value, implying that the ocean dynamics play a role in cooling the climatological mean SST in the EEP, offsetting the net heat flux warming effect. Although the linear ODH term D ic displays a strong interannual oscillation in association with ENSO growth and decay, their net impact on the climatological mean state is near zero. This is understandable because the linear terms contain the product of the mean and perturbation fields, and a time average of such a product must vanish according to the definition. The nonlinear ODH term D ii amplitude is much smaller than the linear counterpart, but interestingly, it keeps the same sign during the El Niño and La Niña years (Fig. 5b, c). Table 3 lists the period-mean values of individual ODH terms averaged in the EEP during the ENSO and NOML years from both ORAS5 and SODAv2.2.4. Note that both the Fig. 4 Horizontal patterns of (the first column) SST (°C, from ERSSTv5), (the second column) surface net shortwave radiation (shaded, W/m 2 ) and total cloud cover (contour, %), (the third column) surface downward longwave radiation (shaded, W/m 2 ) and total cloud cover (contour, %) (Radiation and total cloud cover datasets are from ERA5), and (the fourth column) oceanic dynamic heating (°C/month, from ORAS5) for El Niño, La Niña, ENSO and NOML year composites linear and nonlinear ODH terms ( D ic and D ii ) are near zero, implying that their contributions to the climatological mean SST are negligible. And for nonlinear oceanic dynamic D ii , specifically in three directions, zonal, meridional, and vertical advections are all very small that close to zero (not shown). The contribution from the interesting term D rs is also small. The main contributor is the climatological component ( D cc ).
To examine carefully how ENSO-related linear and nonlinear ODH terms evolve with time, we plotted the composite evolutions of D ii and D ic for El Niño and La Niña, respectively (Fig. 5b, c). It is evident that the amplitude of the linear heating D ic is much greater than the nonlinear heating D ii for both El Niño and La Niña evolutions. The signs of the linear heating D ic are opposite between El Niño and La Niña, as expected. The nonlinear heating D ii has the same sign during El Niño and La Niña, and therefore it might potentially affect the mean SST. However, the 12-month average of D ii is one order of magnitude smaller than that of D ic for both El Niño and La Niña, indicating that the amplitude of nonlinear dynamic heating is too small to make a significant impact.
The horizontal patterns of the linear and nonlinear ODH fields during the ENSO developing phase (from August to October) are shown in Fig. 6. As one can see, a strong warming (cooling) tendency of D ic occurs during El Niño (La Niña) over most parts of the EEP (Fig. 6a, c), with a pattern Fig. 5 a Time series of Dtt (total ODH; black), Dii (nonlinear ODH; red), Dic (linear ODH; blue), Dcc (climatological annual cycle ODH; green), and Drs (the residual; yellow) averaged in the EEP region from 1958 to 2010. b-c Composite temporal evolutions of Dii (solid red) and Dic (solid blue) for El Niño and La Niña, with red and blue dash lines, stand for the 12-month average. The ODH unit is °C/month Table 3 Period- correlation of -0.97 over the EEP region (denoted by the black box). This contrasts with the D ii field, which shows a similar pattern between El Niño and La Niña (Fig. 6b, d), with a pattern correlation of 0.68 over the concerned region, indicating they have almost the same contributions. Notably, a warming tendency appears in the far EEP, consistent with Su et al. (2010). A negative nonlinear ODH appears in the western part of the EEP box, and it offsets the positive tendency to the east, leading to an overall negligible contribution.

ENSO impact on the mean state: idealized model experiments
In the previous section, by separating observed records into an ENSO state and a non-ENSO state, we investigate the possible effect of ENSO on the mean state by diagnosing an equilibrium mixed-layer heat budget model. This section further examines the ENSO impact through idealized numerical modeling experiments. Four experiments are designed. In the CTRL run, the OGCM is forced by observed daily forcing fields for 1958-2010. Figure 7a shows the simulated Niño-3.4 index from this run. The model can capture the observed SSTA evolution in the region. The correlation coefficient between the simulated and observed time series is 0.94. In EXP_ AM + IA, as shown in Fig. 7b, the SSTA evolution has a correlation coefficient of 0.92 with that in CTRL, indicating their robust relationship and the ability of this experiment to capture the realistic observed interannual SSTA variability. The realistic simulation of the interannual SSTA variability adds the confidence to conduct a sensitivity experiment EXP_NO_IA, which is forced by idealized daily forcing fields with their interannual variability removed by Butterworth 1 to 8-year band-pass filter, and EXP_AM with long-term mean atmospheric forcing only to further reveal the role of ENSO on the mean state. As seen in Fig. 7c, the SSTA variability in the Niño-3.4 region in EXP_NO_IA is greatly suppressed, and so is in EXP_AM in Fig. 7d. The correlation coefficient of the Niño-3.4 index time series between EXP_NO_IA (EXP_AM) and the model observed one in CTRL is 0.19 (− 0.15), indicating that they are unrelated. The interannual SSTA standard deviation distribution in the CTRL (Fig. 7e) resembles the observed, with a maximum SSTA variability center located at the EEP, while for EXP_AM + IA, the maximum center appears in the far eastern Pacific but with a more extension to the west near the date line (Fig. 7f). As for EXP_NO_IA and EXP_AM, however, due to the exclusion of interannual signals, the Given the distinctive contrasts of the interannual SSTA variabilities between the CTRL and EXP_NO_IA, and EXP_AM + IA and EXP_AM, we are curious whether there would be a significant difference in the mean climate state. Figure 8a shows the mean SST (solid black) difference along the equator between the CTRL and EXP_NO_IA. As one can see, the SST difference is tiny over the EEP. This confirms the observational analysis that ENSO variability has little impact on modulating the tropical Pacific's mean state. Applying the same diagnosis framework with Eq. (10), one may analyze the relative contributions of the dynamic and thermodynamic terms. The difference operator is between the model and observational diagnoses in the previous section. Instead of diagnosing the difference between the ENSO and non-ENSO years, we now calculate the difference between the CTRL and EXP_NO_IA for the entire model integration period. The longitudinal distributions show that the net effects of the ocean dynamics and the surface heat fluxes are minor along the equator. In comparison, the long-term mean SST difference (solid black) between EXP_AM + IA and EXP_AM and the individual terms of dynamic and thermodynamic contributions along the equator in Fig. 8b depicts the same result, adding more confidence to the weak modulation of ENSO on the mean climate state. Moreover, following the ENSO event year definition applied in the previous section, ENSO and NOML years are separated in the CTRL, and their periodmean SST difference (solid black) is also close to zero in the EEP region, with the contributions of each dynamic and thermodynamic term resembling the observed. The numerical result generally agrees with the observational analysis result shown in Sect. 3.
Following that method applied in the analysis in Sect. 3, we split the ODH into nonlinear heating D ii , linear heating D ic , climatological annual cycle heating D cc , and the residual D rs . Figure 9a-d show the time evolutions of the total ODH and the decomposed individual ODH components in the CTRL, EXP_AM + IA, EXP_NO_IA, and EXP_AM respectively. In all four experiments, the total ODH fluctuates around the climatological annual cycle component D cc with a negative time-mean value that indicates an overall dynamic cooling effect for the mean state. The linear Fig. 7 Time series of NINO3.4 from 1958 to 2010 in a CTRL (red) and observation (from ERSSTv5, black) with correlation coefficient as 0.94, b EXP_AM + IA (red) and CTRL (black) with correlation coefficient as 0.92, c EXP_NO_IA (red) and CTRL (black), and d EXP_AM (red) and CTRL (black) (the correlation coefficients all passed 95% significance test). And standard deviation distributions of the 1 to 8-year band-pass filtered SSTA in e CTRL and f EXP_ AM + IA, with a black box denoting the EEP region ODH term D ic , on the other hand, shows a great difference between the CTRL and EXP_NO_IA, and so does between EXP_AM + IA and EXP_AM. This is consistent with strong (weak) ENSO variability in the CTRL and EXP_AM + IA (EXP_NO_IA and EXP_AM). Due to the lack of ENSO variability, the linear and nonlinear ODH terms are extremely small in EXP_NO_IA and EXP_AM.
To illustrate the impact of ENSO on the ODH more clearly, we show the differences of each of the ODH terms between the CTRL and EXP_NO_IA in Fig. 9e and between EXP_AM + IA and EXP_AM in Fig. 9f. As expected, the climatological ODH is identical in the two experiments, so that their difference is zero. The most considerable difference arises from the linear ODH term, followed by the nonlinear ODH term.

Summary
Motivated by a previous theoretical study that suggested that nonlinear ocean dynamic heating associated with ENSO might significantly affect the mean SST, we investigated the effect of ENSO rectification processes on the Pacific mean climate through a combined observational and modeling study. A simple analytical model was developed to quantitatively delineate the contributions of oceanic dynamic heating and surface heat fluxes to the rectification effect. This methodology follows the idea of Zhang and Li (2014), who studied the formation of future Pacific mean SST pattern under global warming.
A strategy was developed to reveal the ENSO impact on the time-mean state by introducing two equilibrium states during 1958-2010. There is a pronounced interannual variability in the tropical Pacific in the first equilibrium state (i.e., the ENSO state). On the other hand, there is no ENSO variability in the second equilibrium state (i.e., normal state). By diagnosing differences in the mean SST and associated dynamic and thermodynamic processes between the two states under the constructed theoretical framework using the observational data, we unvain the possible ENSO impact on the mean state and specific processes contributing to the rectification effect.
Our observational diagnosis indicates that the observed differences in the mean SST between the ENSO and non-ENSO states are minor. Despite the distinctive interannual variabilities between the two states, ENSO's overall dynamic Fig. 8 Longitudinal distributions of simulated (solid black) and estimated (dash black, based on Eq. 10) a long-term mean SST differences between CTRL and EXP_NO_IA, b long-term mean SST differences between EXP_AM + IA and EXP_AM, and c period-mean SST differences between ENSO and NOML years in CTRL, along the equator (4°S ~ 4°N). Colored lines show the contributions from five individual terms in Eq. (10) and thermodynamic rectification effects on the mean SST in the tropical Pacific are too weak to make a significant difference. The constructed quasi-equilibrium mixed-layer heat budget framework is further used to estimate the impact of each dynamic and thermodynamic term quantitatively. The result shows that the effects of surface heat flux processes are weak. Surface downward longwave radiative heating tends to offset the net shortwave radiative heating. Even though linear ocean dynamic heating terms are strong, they tend to cancel out by ENSO warm and cold phases. On the other hand, the nonlinear ocean dynamic heating terms have the same sign during El Niño and La Niña, but their contribution to the mean SST is weak due to their weak amplitude.
Idealized OGCM experiments are further designed to confirm the observational diagnosis result. In a control experiment, the model is forced by observed daily atmospheric forcing fields. And the model can reproduce the observed ENSO variability in the tropical Pacific. Three experiments are further conducted, in one of which daily atmospheric fields force the model with the interannual signal being removed (EXP_NO_IA), and two other experiments are designed to make a further comparison. In one of them, the long-term mean atmospheric forcing of 1958-2010 (EXP_AM) is applied, while in the other, the atmospheric forcing used is the long-term mean atmospheric forcing of 1958-2010 plus the interannual signals over the period (EXP_AM + IA). Despite the great contrast in the ENSO variability, the experiments reproduce an almost identical long-term mean SST field, implying that the ENSO rectification effect on the mean state is small. Using the same Fig. 9 Time series of Dtt (total ODH; black), Dii (nonlinear ODH; red), Dic (linear ODH; blue), Dcc (climatological annual cycle ODH; green), and Drs (the rest; yellow) averaged in the EEP region from 1958 to 2010 in a CTRL, b EXP_AM + IA, c EXP_NO_IA, and d EXP_AM. e is the difference between (a) and (c). f is the difference between (b) and (d). The unit for ODH is ℃/month mixed-layer budget diagnosis framework for the model outputs, we found that the rectification effect of atmospheric surface heat fluxes is weak, similar to the observational result. While the linear ODH shows a significant difference, as expected, its accumulated impact on the long-term mean SST is minor. Although the nonlinear ODH has the same sign during El Niño and La Niña events, its accumulated effect on the mean SST is small due to its extremely weak amplitude.
To sum up, our observational and numerical modeling studies demonstrate that the nonlinear rectification effect of ENSO, in particular, the impact of the nonlinear ODH, on the mean SST in the eastern equatorial Pacific is small and insignificant, despite strong interannual variability in situ. The strong ENSO modulation seen in a simple theoretical model (Liang et al. 2012) possibly results from the oversimplification of the model, including the treatment of atmospheric wind responses to SST, surface heat fluxes, and oceanic horizontal and vertical advections.