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To improve the model, it is necessary to understand the physical origins of the errors in simulating the behaviors of ENSO. It has been shown that CAMSCSM overestimates the amplitude of ENSO (Figs. 5, 6). The BJindex, proposed by Jin et al. (2006), provides a powerful dynamical estimation of ENSO’s growth rate (Kim and Jin, 2011; Kim S. T. et al., 2014a), and thus it is adopted here to diagnose the origin of this amplitude bias in CAMSCSM. Based on the recharge–discharge framework, the BJindex can be formulated as follows:
$$\hspace{170pt} {I_{\rm BJ}} = \dfrac{{R  \varepsilon }}{2} \approx \dfrac{R}{2}, $$ (1) $$\begin{array}{l} R \!=\! {\mu _a}{\beta _u} < \!\! \dfrac{{\partial \overline T }}{{\partial x}} > \!+\! {\mu _a}{\beta _h}{a_h} < \dfrac{{H(\bar w)\bar w}}{{{H_{\rm m}}}} > \!+\! {\mu _a}{\beta _w} < H(\bar w)\dfrac{{\partial \overline T }}{{\partial z}} > \\ \;\;\;\;\;\;\;\;  ({a_1}\dfrac{{ < \Delta \overline u > }}{{{L_x}}} + {a_2}\dfrac{{ < \Delta \overline v > }}{{{L_y}}})  {\alpha _{\rm s}}, \end{array}$$ (2) where T denotes ocean temperature averaged over the mixed layer; u, v, and w represent the zonal, meridional, and vertical ocean current, respectively; L_{x} and L_{y} are the zonal and meridional extents of the Pacific; H_{m} is the mixedlayer depth, which is fixed at 50 m for simplicity; a_{1} (a_{2}) is derived by regressing the SST anomalies at zonal (meridional) boundaries against the areaaveraged SST anomalies; H(x) is a step function, which only considers regions with upward vertical advection; the sign “< >” denotes the average of quantities over the eastern Pacific (5°S–5°N, 170°–80°W); and the overbars represent the climatological mean state. For more details about the formulation of the BJindex, readers are referred to Jin et al. (2006).
To obtain the BJindex in Eq. (2), several balance relationships are applied. The parameter
${\mu _a}$ indicates the strength of the wind stress response to ENSO SST forcing;${\beta _u}$ measures the winddriven zonal current anomalies;${\beta _h}$ represents the response of the thermocline slope to the surface wind stress anomalies;${\beta _w}$ measures the winddriven upwelling anomalies;${\alpha _s}$ indicates the surface heat flux changes in response to ENSO SST forcing; and a_{h} denotes the relationship between the subsurface temperature anomaly and thermocline depth anomaly. As demonstrated in Fig. 8, these balance relationships hold relatively well for both the GODAS reanalysis data and the CAMSCSM simulation, which ensures the robustness of the BJindex in representing ENSO’s growth rate.Figure 8. Scatterplots of (a, g)
$[{\tau _x}] = {\mu ^*}_a < T > $ , (b, h)$ < {T_{\rm sub}} > = {a_h} < h > $ , (c, i)$ < w > =  {\beta _w}[{\tau _x}]$ , (d, j)$ < Q > =  {\alpha _s} < T > $ , (e, k)$ < h >  [h] = {\beta _h}[{\tau _x}]$ , and (f, l)$ < u >  {\beta _{uh}}[h] = {\beta _u}[\tau ]$ , in (a–f) observation and (g–l) the CAMSCSM simulation. The linear fitting lines are indicated by the red straight lines. Please refer to Jin et al. (2006) for more details about these balance equations.In Eq. (2), the BJindex consists of three positive feedbacks (zonal advective feedback:
${\mu _a}{\beta _u} <  \dfrac{{\partial \overline T }}{{\partial x}} > $ ; thermocline feedback:${\mu _a}{\beta _h}{a_h} < \dfrac{{H(\bar w)\bar w}}{{{H_{\rm m}}}} > $ ; and Ekman feedback:${\mu _a}{\beta _w} < H(\bar w)\dfrac{{\partial \overline T }}{{\partial z}} > $ ) and two negative feedbacks (mean advection damping:$  ({a_1}\dfrac{{ < \Delta \overline u > }}{{{L_x}}} + {a_2}\dfrac{{ < \Delta \overline v > }}{{{L_y}}})$ ; and thermal damping:$  {\alpha _{\rm s}}$ ). Figure 9 compares each contributing processes of the BJindex in the GODAS reanalysis data and the CAMSCSM simulation. Clearly, the thermocline feedback is the dominant term among all three feedbacks. CAMSCSM tends to produce a weaker thermocline feedback than observed (Fig. 9). This underestimation of thermocline feedback can be attributed to three physical processes. First, the wind stress response to ENSO SST forcing (${\mu _a}$ ) is too weak in the CAMSCSM simulation. Figure 10 demonstrates the regression patterns of precipitation and surface wind onto Niño3.4 anomalies. It can be seen that the ENSOinduced atmospheric convection is fairly weak in the CAMSCSM simulation and, as a consequence, the westerly wind response over the central to western Pacific is underestimated. Second, the thermocline response to surface wind stress is also weak in CAMSCSM. As shown in Fig. 11a, the westerly wind stress tends to lift the thermocline in the western Pacific and deepen the thermocline in the eastern Pacific. However, the simulated thermocline changes are weaker than observed (Fig. 11b), which gives rise to parameter${\beta _h}$ being smaller. Third, the modeled mean upwelling in the central to eastern Pacific (4.2 × 10^{–6} m s^{–1}) is also slightly weaker than observed (4.6 × 10^{–6} m s^{–1}). Above all, in the simulation by CAMSCSM, the ENSOSSTinduced weak westerly wind stress can further induce weak subsurface temperature changes, which are then brought to the mixed layer by the weak mean upwelling. As a result, the modeled thermocline feedback is weaker than observed.Figure 9. The BJindex (BJ) and its five contributing terms (MA: mean advective damping, ZA: zonal advective feedback, TH: thermocline feedback, EK: Ekman feedback, and TD: thermal damping) in observation (blue bars) and the CAMSCSM simulation (red bars). The calculation of the BJindex is given in Eqs. (1) and (2).
Figure 10. Regression patterns of the anomalous precipitation (color shaded; mm day^{–1} °C^{–1}) and surface wind (vectors; m s^{–1} °C^{–1}) against Niño3.4 anomalies in (a) observation and (b) the CAMSCSM simulation.
Figure 11. Regression patterns of anomalous thermocline depth (m Pa^{–1}) against the surface wind stress anomalies along the equator (5°S–5°N, 140°–80°W) in (a) observation and (b) the CAMSCSM simulation.
Despite the underestimation of thermocline feedback, the modeled amplitude of ENSO is still larger than observed. As shown in Fig. 9, the overestimation of negative heat flux damping is remarkable, which overwhelms the underestimated thermocline feedback. The surface heat flux can be divided into four components: shortwave radiation flux, longwave radiation flux, latent heat flux, and sensible heat flux. As shown in Fig. 12, both the shortwave radiation and latent heat fluxes are quite important in ENSO dynamics, which is consistent with previous studies (Lloyd et al., 2011, 2012). The El Niñoinduced reduction in shortwave radiation is remarkably underestimated by CAMSCSM (Fig. 12). The modeled negative shortwave radiation feedback is only one third of the observed value. This underestimation of shortwave feedback prevails in many CGCMs (Sun et al., 2009; Lloyd et al., 2012; Chen L. et al., 2013; Chen and Yu, 2014). Previous studies have pointed out that an excessive equatorial CT is a key factor causing this bias (Chen and Yu, 2014; Chen et al., 2016b). Those models with a more excessive CT will produce a westward shift of the convective response to ENSO, which leads to the underestimated shortwave feedback over the central to eastern Pacific (Chen L. et al., 2013). Here, we show that CAMSCSM also suffers from this common bias in CGCMs. Figure 13 demonstrates the regression patterns of shortwave radiation flux against Niño3.4 anomalies. A pronounced reduction in shortwave radiation is evident from 160°E to 100°W in observation, which cools the El Niñorelated SST anomalies. However, the modeled shortwave reduction is much weaker than observed. In addition, its zonal position displaces in the western Pacific, which cannot reduce the El Niño SST warming in the central to eastern Pacific. Based on a previous argument (Chen L. et al., 2013, 2016b; Chen and Yu, 2014), this westward shift of the atmospheric response is consistent with the excessive CT in CAMSCSM (Fig. 1). In addition, the shortwave radiation even slightly increases in the equatorial eastern Pacific (Fig. 13b). This implies that the modeled atmosphere in the eastern Pacific is quite stable, such that higher SSTs destabilize the boundary layer and prevent the formation of stratiform cloud (Philander et al., 1996). This positive cloud–radiation–SST feedback seems to be dominant over the eastern Pacific in CAMSCSM, which could further increase the downward shortwave radiation and intensify the growth rate of El Niño.
Figure 12. Regression of anomalous shortwave radiation (SW), longwave radiation (LW), sensible heat flux (SH), and latent heat flux (LH) against the SST anomaly in the Niño3.4 region in observation (blue bars) and the CAMSCSM simulation (red bars).
Figure 13. Regression patterns of anomalous shortwave radiation flux (W m^{–2} °C^{–1}) against the Niño3.4 index in (a) observation and (b) the CAMSCSM simulation.
As shown in Fig. 9, the gross effect, as indicated by the BJindex, is negative in observation, which is consistent with the theory that ENSO is a damped oscillation triggered by stochastic forcing (e.g. Kleeman and Moore, 1997; Kirtman and Schopf, 1998). In the simulation by CAMSCSM, the BJindex is also negative but the absolute value is smaller. This implies that the modeled ENSO is a less damped mode, which explains the overestimation of ENSO’s amplitude (Fig. 6). Here, we show that the BJindex is useful in diagnosing the physical origin of the bias in ENSO’s amplitude, especially when error compensation occurs. For CAMSCSM, the weak negative shortwave radiation feedback overwhelms the weak thermocline feedback, giving rise to a stronger modeled ENSO.

Realistically simulating ENSO’s periodicity remains a challenging issue in current CGCMs (Bellenger et al., 2014; Lu and Ren, 2016). For CAMSCSM, the modeled ENSO oscillates regularly with a period of 2.8 yr, which is shorter than observed. Here, the socalled Wyrtki index, proposed by Lu et al. (2018), is adopted to estimate the physical origin of this periodicity bias. Similar to the BJindex, the Wyrtki index is also derived from the linear recharge–discharge framework:
$$\dfrac{{{\rm d} < \!T \! > }}{{{\rm d}t}} = R < \!T\! > + F [h], $$ (3) $$\!\!\!\!\!\!\!\! \dfrac{{{\rm d}[h]}}{{{\rm d}t}} =  \varepsilon [h]  B < \!T \! >, $$ (4) where T denotes SST; h represents thermocline depth; the sign “< >” denotes the average of quantities over the eastern Pacific (5°S–5°N, 170°–80°W); and the sign “[ ]” denotes the average of quantities over the entire equatorial Pacific domain (5°S–5°N, 140°E–80°W); R indicates the damping rate of eastern Pacific SST, and is often referred to as the Bjerknes instability index (Jin et al., 2006);
$\varepsilon $ denotes the damping rate of thermocline depth due to the energy leak at the western boundary and mixing; F represents the impact of the discharged or recharged state of the ocean heat content on SST; and B indicates the efficiency of the discharging or recharging of the equatorial heat content driven by the equatorial wind stress curl anomalies induced by the anomalies of ENSO SST. The noise forcing and nonlinear terms are omitted here for brevity. This simple linear framework is capable of representing the linear lowfrequency dynamics of ENSO’s growth rate (Jin et al., 2006) and frequency (Lu et al., 2018). The Wyrtki index can be formulated as:$${I_{\rm Wyrtki}} = \dfrac{{4\pi }}{{\sqrt {4B \cdot F  {{(R + \varepsilon)}^2}} }} \approx \dfrac{{2\pi }}{{\sqrt {B \cdot F} }}.$$ (5) The simulated parameter B is 3.9 × 10^{–7} m (s K)^{–1}, which is quite close to the observed value of 4.3 × 10^{–7} m (s K)^{–1}. However, the modeled parameter F is 1.3 × 10^{–8} K (s m)^{–1}, which is much larger than the observed value of 8.3 × 10^{–9} K (s m)^{–1}. As a result, the modeled Wyrtki index is 2.8 yr, which is slightly shorter than the observed Wyrtki index of 3.4 yr. In other words, the phase transition of ENSO induced by the thermocline and zonal advective feedbacks is quicker than observed, which contributes to the shorter period of ENSO in CAMSCSM.

In addition to the internal dynamics of ENSO as discussed above, assessing the external forcings for ENSO is also very important, given the fact that ENSO is a damped oscillation mode (Fig. 14) and can be triggered by several precursors, such as North Pacific Oscillation (NPO; Vimont et al., 2001; Alexander et al., 2010), the Indian Ocean Dipole (IOD; Izumo et al., 2010), Arctic Oscillation in spring (Chen S. F. et al., 2014), an anomalous EAWM (Li, 1990, 1996), thermocline changes in the western Pacific (Chen et al., 2016a), and the consecutive occurrence of westerly wind bursts (Chen et al., 2017). We begin by evaluating the precursor of NPO during the previous winter.
Figure 14. Seasonal composites of negativeminuspositive SLP index cases for SST (color shaded; °C) and 10m winds (arrows; m s^{–1}) in observation (lefthand panels) and the CAMSCSM simulation (righthand panels).
As shown in Fig. 14a, the sea level pressure (SLP) changes (negative minus positive) in North Pacific might reduce the trade winds, which in turn may induce footprintlike SST anomalies in the subtropics via latent heat flux changes. This footprint signal persists throughout the spring and potentially affects the tropical Pacific throughout the following year (Figs. 14c, e, f; Vimont et al., 2001). This socalled “seasonal footprinting mechanism” is reproduced well by CAMSCSM, in terms of the seasonal evolution of SST and surface wind (Fig. 14). In fact, the SLP anomalies in North Pacific (10°–25°N, 175°–145°W) during the previous winter are significantly correlated with ENSO (Fig. 15a). CAMSCSM tends to simulate a similar NPO–ENSO relationship as observed (Fig. 15b), implying a realistic seasonal footprinting mechanism in this model.
Figure 15. Scatterplots between the normalized Niño3.4 index averaged during November (0) to January (1) and the normalized SLP index averaged during November (–1) to March (0) from (a) observation and (b) the CAMSCSM simulation. The correlation coefficients are given in parentheses on top of each panel.
Izumo et al. (2010) pointed out that the IOD is another precursor of El Niño, at a lead time of 14 months. Figure 16 compares the lead–lag correlations between the IOD [averaged from September to October of Year (0)] and the Niño3.4 index in observation [1980–2016, following Izumo et al. (2010)] and in the CAMSCSM simulation, separately. A significant positive correlation is evident in Year (0). The El Niño condition is often accompanied by a positive IOD phase, with a strengthening of the easterly winds off Sumatra in summer, which may in turn aid the development of El Niño (Annamalai et al., 2005; Luo et al., 2010). This covariability of ENSO and the IOD is simulated well by the CAMSCSM, with a positive correlation (0.6) in Year (0) (Fig. 16). In addition, a significant negative correlation is clear one year after the IOD peak. In other words, a positive (negative) IOD phase leads the ENSO peak by around 14 months. This relationship has been applied in an ENSO prediction model with reliable skill (Izumo et al., 2010). As shown in Fig. 16b, such a relation of the prior IOD signal with ENSO is also captured by CAMSCSM.
Figure 16. Lagcorrelation between the IOD [averaged from September to November of Year (0)] and the Niño3.4 index (threemonth running average applied) in (a) observation and (b) the CAMSCSM simulation. The dashed lines indicate the 95% confidence level. The vertical dashed lines denote the September–October–November in Year (0) for a cooccurring IOD.