## 1. Introduction

[2] Considerable effort has been invested in understanding and simulating the phenomenon of El Niño-Southern Oscillation (ENSO) by using models of varying complexity. These models range from theoretical ones [*Wang and Fang*, 1996; *Jin*, 1997a, 1997b; *Wang et al.*, 1999; *Wang*, 2001], through the so-called intermediate coupled model [*McCreary and Anderson*, 1991; *Cane et al.*, 1986; *Suarez and Schopf*, 1988; *Battisti and Hirst*, 1989; *Philander*, 1990; *McCreary and Anderson*, 1991; *Kleeman et al.*, 1995; *Picaut and Delcroix*, 1995; *Neelin et al.*, 1998], to complex coupled general circulation models (CGCMs). Both intermediate coupled models [e.g., *Zebiak and Cane*, 1987] and CGCMs have been used to forecast ENSO. Recently, the climate forecast system at the National Center for Environmental Prediction [*Saha et al.*, 2006], the seasonal forecast systems at the European Center for Medium-Range Weather Forecasts, and the Multimodel Ensemble System at EU [*Palmer et al.*, 2004] and at Asia-Pacific Economic Cooperation Climate Center have also been developed for seasonal to interannual climate prediction.

[3] A detailed comparison of ENSO models was given by *Kirtman et al.* [2002]. They indicated that it is difficult to tell which model shows higher forecast capability for the dynamical and statistical models or intermediate and complex models. To improve ENSO forecast skill, it is necessary to explore the fundamental physics of ENSO for which a theoretical model is useful. *Wang and Fang* [1996], *Jin* [1997a, 1997b], and others have developed analytical models to advance the understanding on ENSO physics and have obtained significant results.

[4] “Spring predictability barrier” (SPB) is a well-known characteristic of ENSO forecasts. The SPB is referred to a phenomenon that most ENSO prediction models often experience an apparent drop in prediction skill across April and May [*Webster and Yang*, 1992]. SPBs exist in coupled and statistical models. In some occasions, the SPB is even stronger in statistical models than in GCMs [*van Oldenborgh et al.*, 2005]. Many works have investigated this phenomenon [*Walker*, 1924; *Webster and Yang*, 1992; *Webster*, 1995; *Moore and Kleeman*, 1996; *Samelson and Tziperman*, 2001; *McPhaden*, 2003, etc.], but debate remains concerning its cause. One of the possible causes is the rapid seasonal transition of monsoon circulation during the boreal spring that perturbs the Pacific basic state when the east-west sea surface temperature (SST) gradient is the weakest [*Webster and Yang*, 1992; *Lau and Yang*, 1996]. Another notion proposed by *Webster* [1995] is that SPB is due to the weakest ocean-atmosphere coupling during spring in the eastern Pacific. Other studies argued that SST anomalies in boreal spring are relatively small, such that these anomalies are difficult to be detected and forecasted in the presence of atmospheric and oceanic noises [*Xue et al.*, 1997; *Chen et al.*, 1995]. *Samelson and Tziperman* [2001] demonstrated that SPB is an inherent characteristic of ENSO, whereas *Chen et al.* [1995, 2004] suggested that this predictability barrier could be reduced through improving initialization. *McPhaden* [2003] showed that subsurface information has a winter persistence barrier and that the predictability of ENSO bestriding spring can be greatly enhanced by incorporating this information into the model. In general, the cause of the SPB remains elusive. There is an urgent need to further address the problems related to SPB for ENSO.

[5] Understanding of SPB can be gained by studying the initial error growth. *Moore and Kleeman* [1996] and *Samelson and Tziperman* [2001] have investigated the seasonal variability of ENSO error growth and explored the cause of predictability barrier by using linear singular vector (LSV). *Blumenthal* [1991], *Xue et al.* [1997], *Thompson* [1998], and *Moore and Kleeman* [1999] also used LSV to study ENSO predictability. LSV is one of the useful tools in predictability studies, but it deals with sufficiently small initial perturbations and thus is unable to describe nonlinear evolution of finite-amplitude initial perturbations [*Oortwijin and Barkmeijer*, 1995; *Mu et al.*, 2003].

[6] In this paper, we use conditional nonlinear optimal perturbation (CNOP) proposed by *Mu et al.* [2003] (see also *Mu and Zhang*, 2006) to study seasonal reliance of the evolution of finite-amplitude initial errors. The CNOP represents the initial error that has the biggest effect on the forecast results at the prediction time. CNOP method has been applied to study the ENSO precursor [*Duan et al.*, 2004] and the sensitivity of ocean thermohaline circulation to finite-amplitude perturbations [*Mu et al.*, 2004], as well as the passive variability of the thermohaline circulation [*Sun et al.*, 2005]. The aforementioned studies illustrate that CNOP is one of the useful tools for predictability study. A particular appealing aspect is that CNOP describes the optimal initial perturbation of nonlinear model and reveals the effect of nonlinearity on predictability.

[7] To facilitate theoretical understanding of essential dynamics and physics for ENSO predictability, we may apply CNOP method to a nonlinear ENSO model. The theoretical one developed by *Wang and Fang* [1996] may provide a convenient tool for analyzing the essential physics of SPB. *Burgers et al.* [2005] revealed SPB phenomenon by using the simplest ENSO recharge oscillator model advanced by *Jin* [1997a, 1997b], which is linear and consists of two-variable ordinary differential equations with stochastic forcing. The model of *Wang and Fang* [1996] (hereinafter referred to as WF96) is a nonlinear ENSO model of two-variable differential equations with self-sustained oscillation that may also capture the essential physical of SPB. In this research, we will use WF96 model to investigate SPB with CNOP approach.

[8] We briefly review the ideas of CNOP in the next section and introduce the ENSO model in section 3. Seasonality of the initial error growth for ENSO is examined in section 4. In section 5, we discuss the mechanism of the seasonal variation for error growth and explain the role of nonlinearity in error growth. We summarize the conclusions in section 6 and finally discuss the implication of our results in section 7.