1. Introduction

Seasonal climate prediction using dynamical models has gained great importance over the last decade. The success of seasonal dynamic prediction hinges strongly on the strength of atmospheric responses to the anomalous lower boundary conditions, particularly the sea surface temperature (SST) over the tropical Pacific Ocean (Shukla et al. 2000 and references therein). Short-term (1–2 weeks) dynamic predictions are typically made by running an atmospheric general circulation model (AGCM) forced with the current state of SST, sea ice, and land surface conditions. However, for a climate prediction with a lead time longer than two weeks, it is necessary to estimate the future state of the lower boundary conditions, particularly the SST field. The first successful dynamical prediction of the El Niño–Southern Oscillation (ENSO), which explains most of interannual SST variability rooted in the tropical Pacific, was done with the Zebiak–Cane model (ZC; Zebiak and Cane 1987). Nowadays, several operational forecast centers around the world use coupled atmosphere–ocean general circulation models (GCMs) to forecast El Niño and its global impact with a reasonable skill up to several months in advance (more information available online at http://www.iges.org/ellfb).

Most operational centers adopt a so-called two-tiered system in making seasonal climate predictions. In the first tier (or step), the future SST anomalies are forecasted by running a coupled ocean–atmosphere GCM. In the second tier, the forecast SST serves as the lower boundary forcing over the oceans for an ensemble of atmospheric GCM forecasts [the Bengtsson et al. (1993) and Palmer and Anderson (1994) experiments in Europe; Ji et al. (1996), Livezey et al. (1997), and Barnston et al. (1999) at the National Centers for Environmental Prediction (NCEP); Mason et al. (1999) at the International Research Institute for Climate Prediction (IRI); and Chang et al. (2000) and Pegion et al. (2000) in the National Aeronautics and Space Administration (NASA) Seasonal-to-Intraseasonal Prediction Project (NSIPP)]. As Toth and Kalnay (1996) pointed out, this two-tier approach suffers from an important deficiency. Since it uses a single forecast for the SST, it neglects the uncertainty in the SST forecast, which is central to the initial uncertainty in the ENSO phenomena that is being forecasted. Experience with atmospheric ensemble forecasting indicates that such an assumption can negatively impact forecast performance, and that the use of a single SST forecast probably reduces skill levels compared to an ensemble mean. Moreover, flow-dependent estimates of forecast uncertainty are severely limited (e.g., Ehrendorfer 1997).

Recently, the European Centre for Medium-Range Weather Forecasts (ECMWF) started to use a “single-stage” configuration of coupled general circulation models (CGCMs) in making ensemble seasonal climate predictions (Stockdale et al. 1998). This is an advanced setup where, unlike in the two-tier system, all ensemble forecast members are computed with the coupled ocean–atmosphere model. Initial perturbations, however, are introduced only in the atmospheric component of the coupled system. Though perturbations through the coupling will eventually appear in the ocean part of the system as well, this approach is still handicapped by the lack of the representation of uncertainty in the initial state of ENSO. Again, experience with atmospheric ensemble forecasting indicates that such an approach has limitations.

Toth and Kalnay (1996) proposed to use the breeding method to generate initial perturbations in a coupled ocean–atmosphere model to represent the initial uncertainty of the coupled system. The ultimate scientific goal of our research within NSIPP is to apply this idea in a fully coupled CGCM system to improve coupled model data assimilation and ensemble forecasting skill for seasonal climate prediction. The purpose of the present paper is to investigate the applicability of the breeding method with the Zebiak–Cane model. We chose the ZC model to investigate the basic ideas and feasibility of the proposed approach. The ZC model is the first coupled atmosphere–ocean model that simulates ENSO variability successfully. The ZC model predicts anomalies about a prescribed seasonally varying background flow. The oceanic component of the ZC model consists of a linear-reduced gravity equatorial β-plane shallow-water model with a simple surface mixed layer and a fully nonlinear SST equation. The atmospheric component is a Gill diagnostic model driven by a heating anomaly associated with both SST anomalies and low-level moisture convergence anomalies. Readers are referred to Chen et al. (2000) and the references therein for updated information about the ZC model. Because the variability associated with weather is explicitly excluded in the ZC model, we would have little ambiguity in relating the bred vectors to the sole interannual variability, namely, ENSO, present in the ZC model. In addition, because of its low order, we can run the ZC model as many times as needed for testing ideas. With the ZC model, we try to accomplish two tasks: 1) understand the characteristics of the bred vectors in relation to the background ENSO variability; 2) test the utility of applying the bred vectors in the context of data assimilation and ensemble forecasts for ENSO predictions within a very simple setup.

Recently, there have been several important papers on ENSO predictability using the ZC model (e.g., Chen et al. 1997; Xue et al. 1997a,b; Thompson 1998; Thompson and Battisti 2000, 2001; Ballabrera-Poy et al. 2001). All of these studies used singular value decomposition to obtain the singular vectors (SVs) of the ZC model (or the Battisti model, a close relative of the ZC model). Moore and Kleeman (1996, 1997a,b) studied the characteristics of SV modes associated with ENSO variability and estimated the impact of the dynamic growing errors on the ENSO predictability with a more realistic intermediate coupled model for the Pacific basin. In this paper, we compare whenever possible the characteristics of the bred vectors with those found in the SV analysis studies.

The second section includes a brief discussion of the breeding method. Section 3 is devoted to the implementation of the breeding method in the ZC model, and the sensitivity of the bred vectors to the choice of breeding parameters. Section 4 describes the characteristics of the bred vectors of the ZC model with an emphasis on their relation to both the background ENSO evolution and the annual cycle. Section 5 illustrates the potential benefits of using bred vectors within a simplified context related to data assimilation and ensemble forecasting, while section 6 summarizes the main conclusions.

2. The breeding method

Toth and Kalnay (1993) proposed a method denoted “breeding of fast-growing modes” for short-to-medium-range atmospheric ensemble forecasts. The fast-growing modes obtained by breeding are known as bred vectors, and they are essentially the finite-time, nonlinear extension of local Lyapunov vectors (Toth and Kalnay 1997). This method was implemented at NCEP (formerly the National Meteorological Center) in late 1992 for its operational 1–15-day ensemble forecasts. Breeding and related methods have been adopted by several research and operational centers partly due to their simplicity and computational efficiency (India, Japan, U.S. Navy, South Africa). Ensemble-based data assimilation techniques developed recently are also closely related to breeding (e.g., Houtekamer and Mitchell 1998; Hamill and Snyder 2000; Bishop 2001, manuscript submitted to Mon. Wea. Rev.).

An alternative way to generate fast-growing perturbations is the SV method and the corresponding perturbations are often referred to as “optimal” perturbations. The SV method has been applied to ensemble forecasting at ECMWF since late 1992 (Molteni and Palmer 1993; Buizza and Palmer 1995; Molteni et al. 1996), and has been tested experimentally by several research groups. As indicated in the introduction, SVs have also been applied to the ZC model, and in this paper we compare several of the BV and SV results. In both the BV and SV methods, it is assumed that the model errors are not the dominant source of errors. The SVs are computed under a linear perturbation assumption.

Here we summarize the essential ingredients of the breeding method. Bred vectors are difference fields between two nonlinear integrations of a dynamic system, started with a small arbitrary initial perturbation. In a dynamical system with at least one expanding direction such difference fields will necessarily project onto the fastest-growing directions. The presence of nonlinearities, however, will ultimately limit growth. In complex systems like the atmosphere or the coupled ocean–atmosphere where various instabilities are present, nonlinear inhibition of growth occurs selectively, first on processes with the lowest saturation amplitude. To avoid or limit nonlinear saturation, and, hence, allow the perturbations to continually span the fastest-growing subspace of the system, the difference fields have to be periodically rescaled. Such a periodic rescaling of nonlinear perturbations is called a breeding cycle (Fig. 1). The resulting bred vectors are closely related to the Lyapunov vectors (LVs). The leading LV is estimated through a procedure very similar to breeding (see, e.g., Benettin et al. 1980), the only difference being that for the LVs the perturbation is carried forward via a linear tangent model, instead of being the difference between two nonlinear integrations. Periodic rescaling is used in LV methods merely to avoid computer overflow resulting from successive perturbation growth, whereas in breeding its purpose is to selectively control which instabilities to “breed,” and which to saturate.

Breeding is controlled by only two parameters, the size of the initial (and rescaled) perturbation amplitude and the length of time for which the perturbation is let to develop before rescaling it again. As will be shown later on in this paper, bred vectors are practically insensitive to the choice of the norm (see also Toth and Kalnay 1993, 1997). The two parameters of breeding basically determine the range of amplitudes the perturbations can assume. Perturbations related to instabilities with saturation levels below this range will not be represented in the BVs (or they would appear only in their saturated, nonexplosive form). The BVs consist of perturbations related to instabilities whose saturation amplitudes are well above the range of amplitudes allowed and whose growth rate is largest, since perturbations related to slower-growing instabilities are damped more strongly by the rescaling cycle. This is, again, a process analogous to the natural emergence of the first, fastest-growing LV in a linear perturbation environment (see, e.g., Trevisan and Legnani 1995; Boffetta et al. 1998).

Figure 1 is a schematic diagram showing a breeding cycle run upon a control integration. Typically, it takes only a few days of breeding for a random initial perturbation to turn into a BV in the case of an atmospheric GCM. In general the time period needed for an arbitrary perturbation to turn into a dynamically fast-growing direction is related to the growth rate spectrum of fast-growing perturbations (SVs or LVs). If the perturbation amplitudes are kept sufficiently small, the breeding method can also be used to estimate LVs of a nonlinear dynamic system without linearizing the system. Since the breeding method is based on the use of a nonlinear model with full physical parameterizations, when it is applied with finite perturbation amplitudes it naturally accounts for all nonlinear interactions that can be otherwise hard to simulate with the simplified physics usually introduced into linear tangent models corresponding to atmospheric and oceanic processes.

Kalnay and Toth (1994) pointed out that data assimilation cycles in essence are very much analogous to breeding cycles. In the context of data assimilation, the “truth,” which is not observable but is estimated within a certain degree of accuracy, plays the role of the “control run” in the breeding method. The model forecast (first guess) made from the previous analysis corresponds to the “perturbation run” in the breeding method. The introduction of observations into the data assimilation system in the creation of a new analysis is effectively equivalent to resizing the perturbations in a breeding cycle. Although the resizing that takes place in the assimilation of observations has different amplitudes for different horizontal scales and distribution of observations, this does not significantly affect the shape of the dominant errors (Corazza et al. 2002a,b). In light of this, Kalnay and Toth (1994) argued that the errors in the first guess fields should have strong projections on the bred vectors.

Given its computational efficiency and success in operational atmospheric ensemble forecasting, the breeding method would seem a good candidate for generating ensemble perturbations for seasonal to multiseasonal climate predictions with a CGCM system, because it would naturally include instabilities of the evolving coupled flow. The challenge in applying the breeding method is to find a way to separate the ENSO-related coupled variability from the weather modes and other shorter timescale oceanic instabilities. There exists a similar problem in numerical weather forecasting with a GCM, namely, separating synoptic-scale instability from convective instability (Toth and Kalnay 1993). In this case, the two types of instabilities are separated not only by timescale but also by amplitude. As a result, the synoptic-scale instability is easily identifiable with the breeding method by using a relatively large perturbation amplitude. In the context of ENSO prediction, however, the ENSO-related extratropical variability (at least for the atmospheric part if not the oceanic part) is comparable to that of weather-related variability. Therefore, it seems that they may be separated only through their different timescales and not through their amplitudes.

3. Application to the ZC model

Our implementation of the breeding method in the ZC model is straightforward. The control run is a 1010-yr integration of the ZC model. We made 16 perturbed breeding cycles in order to test the sensitivity of the bred vectors to the choices of the breeding parameters mentioned above. Each of the perturbation runs starts at the first month of the 11th year of the control run and lasts 1000 years. Table 1 summarizes the breeding parameters used in the 16 perturbation runs. We use six variables to determine the resizing factor (or perturbation amplitude). They are oceanic currents on the coarse grids of the ZC model, thermocline depth, SST, and atmospheric winds. For the L2 norm, the perturbation amplitude S is defined as

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where V_i is a ZC model variable, belonging to one of the six variables mentioned above. The superscripts “p” and “c” represent the perturbed and control runs, respectively. The overbar denotes a time mean averaging operation and σ_i is the standard deviation of the corresponding variable calculated from the control run, which is defined as

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The integration is done over the ZC model domain. The perturbation amplitude S for the ocean energy norm is defined as

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where u_o and υ_o are the oceanic currents and H is the reference thermocline depth of the ZC model. The overbar denotes a time-average operation. In (3), H is equal to 150 m and g′ is chosen such that = 2.9 m s^–1. Again, the integration is done over the ZC model domain.

The perturbation amplitude S is calculated on the first day of each model month and the differences between the perturbation and control runs are rescaled on the first day of every M₀ months to a fixed perturbation amplitude S₀. During breeding cycles, we also calculate the monthly amplification factor according to

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where, t is time in unit of month. At the rescaling times, the amplification factor is calculated before the perturbation field is rescaled.

We first tested the sensitivity of the bred vectors of the ZC model to choices of parameters for a period of several years. We found that if we used the same initial random perturbations, the bred vectors were essentially independent of the choices of norm, its size, and the period of rescaling. Because of the nonlinear evolution of the bred vectors, the sensitivity to the initial perturbations (beyond a transient period of about 6 months) was small but not negligible.

We then explored the robustness of the bred vectors over a very long period. The results are summarized in Fig. 2, displaying scatter diagrams of the map correlation of the SST field of the bred vectors obtained with two different sets of breeding parameters. There are 12 000 points on each of the diagrams, and the number of bred vectors with a given pattern correlation is shown on the right of the figures. These scatter diagrams reflect their sensitivity to both the choice of parameters and to the nonlinear evolution of the structure of the bred vectors, which, as indicated before, does not reach complete convergence. Since the bred vectors are proxies of LVs, the amplitude comparison of bred vectors obtained with two different sets of breeding parameters are irrelevant, but the spatial patterns are meaningful. The positive phase of the same pattern should be regarded to be equivalent to the negative phase, at least from a linear system point of view. Therefore, the bred vectors obtained with two different breeding parameters can be regarded as “identical” if the spatial correlation is equal to either 1 or −1. As indicated in Fig. 2, the spatial correlation of bred vectors obtained with two different sets of breeding parameters has the largest population with a value exceeding 0.8. Occasionally, the correlation becomes close to −1. About 25% of the points have pattern correlations between −0.6 and 0.6. This implies some sensitivity of the bred vectors to the breeding parameters and to the nonlinear evolution. However, some of these may be due to the frequent transition from the same phase of the bred vector to the opposite phase in two different breeding runs, as discussed in the next section when Fig. 3 is presented. Nevertheless, it is easy to notice that the bred vectors are the least sensitive to the norm used to measure the size of the perturbation (Fig. 2a vs Figs. 2b,c). This lack of sensitivity is in accordance with the notion that bred vectors are a nonlinear generalization of leading Lyapunov vectors, which are independent of the choice of norm.

In the remainder of the paper, results will be presented with the breeding parameters set in bold in Table 1 (i.e., the L2 norm, with initial perturbation amplitude equal to 10% of the ZC model variability, with the 3-month cycling length). We have confirmed that the results obtained with other combinations of the breeding parameters are essentially the same as those presented here.

4. The bred vectors of the ZC model and their relation to the background ENSO

The bred vectors are defined through the same six variables that are used in calculating the perturbation amplitude (i.e., oceanic currents on coarse grids, thermocline depth, SST, and atmospheric winds). Both the orientation (spatial pattern) and growth rate [Eq. (4)] of the BVs are evaluated on the first day of each month over the 1000-yr (12 000 month) experimental period. For display, comparison, and analysis purposes BVs are renormalized so they all have a uniform norm that is arbitrarily set to 1. This will facilitate the use of statistical diagnostic tools, such as composite and EOF methods that will be used to ascertain the relation between the bred vectors and the underlying ENSO cycle. Note that this renormalization neither interferes with the breeding cycle nor changes the amplitude ratios among the different variables in the bred vectors.

Figure 3 shows a 100-yr portion of the time series of the Niño-3 (5°N–5°S, 90°–150°W) index of the control run (solid line) and the bred vectors. The multiyear timescale of somewhat irregular variability of the Niño-3 index of the control run simply is a manifestation of the ZC model’s ability to simulate ENSO events. We performed an EOF analysis that indicates that the bred vectors are dominated by an ENSO-like mode, which explains about 72% of the variability of the bred vectors. The first EOF pattern (not shown) is somewhat spatially similar to the ENSO background, but about 90° out of phase in time. We use the time series of the Niño-3 index of the bred vectors displayed in Fig. 3 [similar to the principal component (PC) of the first EOF] to characterize the temporal variability of the bred vectors spatial structure as the underlying ENSO evolves. Because the displayed bred vectors have been renormalized, a larger Niño-3 perturbation value is indicative of a dominant ENSO structure, while a smaller value indicates spatially less coherent and smaller-scale structures.

It is seen that the bred vector in the ZC model tends to have a spatially more coherent ENSO-mode structure either prior to or after an extreme event. This is particularly true for the warm events. We also point out that the number of months during which the bred vector has a positive Niño-3 index is nearly equal to the number of negative cases. Very often, the bred vector changes its phase, either from a negative to a positive or from a positive to a negative, as the underlying ENSO passes the warmest extreme stage. Occasionally, however, the bred vector stays with the same phase during the transition (e.g., between years 62–65 in Fig. 3). The bred vectors obtained with different breeding parameters very often experience nearly identical temporal evolution. This explains the overwhelmingly large number of points clustered above 0.6 in Fig. 2. The disagreements come when the bred vectors obtained with different breeding parameters occasionally differ (e.g., one alters the phase and the other does not as the underlying ENSO event passes the warmest phase). This accounts for the points that have a map correlation below −0.6 in Fig. 2. As argued in the previous section, some of the points between −0.6 and 0.6 in Fig. 2 can be attributed to the transition from “the same phase” to “the opposite phase” or vice versa of the bred vectors obtained with different breeding parameters.

Literally, there are hundreds of ENSO events during the 1000-yr realization of the control run. To gain a better understanding of the relation between the bred vectors and the underlying ENSO events, we construct composite maps of both the control run and bred vectors. The background ENSO events are binned into 24 categories. The boundaries of the 24 bins are based on the Niño-3 index of the control run and its temporal tendency. Table 2 summarizes the bins’ lower and upper limits in terms of the Niño-3 index and its temporal tendency.

By the choice of the bins, the composite ENSO event has the coldest SST anomaly in bins 1 and 24 and the warmest SST anomaly in bins 12 and 13. Bins 1–3 and 22–24 have a negative Niño-3 index and the remaining bins correspond to a positive Niño-3 index. From bin 1 to bin 12, the composite ENSO event evolves from the cold to the warm phase and the process is reversed from bins 13 to 24.

We next construct composite maps of the bred vectors by averaging all cases within each of the 24 bins. As suggested in the 100-yr sample period shown in Fig. 3, it is nearly equally possible for the bred vector to be in either the positive or negative phase within each of the 24 categories. Therefore, we have made two sets of composite maps of the bred vector for each of the 24 bins, one with a positive (CBV+) and another with a negative Niño-3 index (CBV−).

Figure 4a contains a series of maps displaying the composite ENSO event (a total of 11 maps), corresponding to bin numbers 1, 3, 5, 7, 9, (12 + 13)/2, 16, 18, 20, 22, and 24 from the top to the bottom. The corresponding spatial structures of CBV− and CBV+ are displayed in Figs. 4b,c, respectively. As indicated in Fig. 3, the composite bred vector has a large amplitude during the transitions between the extreme phases of the underlying ENSO events, exhibiting a large-scale spatial pattern somewhat similar to ENSO over much of the equatorial Pacific basin (Figs. 4b,c). Note also that the positive phase of the bred vector is nearly a mirror image of the negative phase, indicating the bred vector is obtained more or less within the linear regime.

Figure 5 displays the composite mean of the monthly amplification factor σ as a function of the background ENSO phase, where σ is calculated according to (4). It is clear that the bred vectors tend to have a larger growth rate between the extreme phases of ENSO events. The composite bred vector has a negative growth rate (σ < 1) just after the warmest phase of the composite ENSO event. The average amplification factor is about 1.3 month^–1. Figure 6 plots the mean monthly amplification factor of the bred vector as a function of the calendar month. It shows that the mean growth rate of the bred vectors in the ZC model has a strong seasonal variability. It reaches the maximum value in the month of July and has a minimum in November with a slightly negative mean growth rate. Between February and May the growth rate is rather constant. In his study, Battisti (1988) included only the annual cycle as background, and found that the maximum growth of ENSO instability with respect to the climatological annual cycle occurs in the months of June and July. Because our growth rate calculation is done for a system that exhibits both annual and ENSO cycles, this agreement suggests that the presence of ENSO variability on top of the annual cycle is not the dominant factor in creating stronger coupled instability over the summer.

It is of interest to compare the seasonal variation of the instability obtained from the breeding method with the results of optimal instability analysis using the SV method, which, unlike breeding, is strongly dependent on both the choice of norm and the period of optimization. As reported in Chen et al. (1997, using the Battisti model), Xue et al. (1997a,b, with the ZC model), and Thompson et al. (2000, with the Battisti model), the optimal instability (growth rate of the leading singular vector) has little seasonal dependence when the period of optimization is 3 months or less. As the optimization period is lengthened, the optimal growth gradually starts to show an increasing seasonal dependence. Moreover, when the optimization period exceeds 3 months, the SV analysis reveals that maximum growth is observed for optimization periods started in spring (March–April–May). The SV results for optimization periods longer than 3 months thus agree with our findings that the growth rates peak during the summer season. The agreement is not surprising since, as discussed by Szunyogh et al. 1997 (see also Toth et al. 1999; Kalnay 2002, and references therein), SVs with long enough optimization periods tend to turn by the end of their optimization period toward the leading LVs of the system that represent the dominant and norm-independent instabilities of a dynamical system. In a nonlinear environment these instabilities are well captured by the BVs. With an SV analysis, however, it is possible to uncover them only with an optimization period of the order of the natural timescale of the unstable processes and only at the end of the optimization period.

To our knowledge, the growth rate as a function of the phase of the ENSO cycle (e.g., Fig. 5) has not been previously systematically studied in long model runs. Because it is expensive to obtain SVs in the presence of ENSO cycles, these calculations were typically carried out only for a time period of about 20 years, effectively covering about four to five ENSO events. Nevertheless, the SV studies of Chen et al. (1997) and Xue et al. (1997a,b) suggest that the optimal modes tend to grow fast prior to the onset or the breakdown of an ENSO event, in agreement with our results.

P. S. Schopf (2000, personal communication) pointed out that the relation between the growth rate of the fast growing modes and the ENSO cycle, as illustrated in Fig. 5, can be readily explained by a simple delayed oscillator model. According to Suarez and Schopf (1988), a delayed oscillator model for ENSO can be written as

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where, T represents the amplitude of the growing perturbation, t is dimensionless time, Δ is the nondimensional delay time, and α measures the amplitude of the delayed signal relative to the linear and the nonlinear feedbacks. If we use T to represent the control run solution of (5) and (T + T^P) as the perturbed solution, we then may write the linear tangent model of (5) for the perturbation as

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It follows that the perturbation T^P has a larger growth rate when the amplitude of the control solution is small (or the background ENSO event is in the neutral state). As observed in Fig. 5, the growth rate of the perturbed solution should reach a minimum value when the background ENSO is in one of its two extreme phases. In more realistic applications, the variations of the amplitude of the ENSO episodes needs to be taken into account.

5. Potential applications

In this section we discuss simple experiments in the context of the perfect model scenario showing the potential impact of using breeding in data assimilation and ensemble forecasting. Although in these experiments we assume that we know the truth, the results should still be representative of the real applications in which the true state of the coupled system is only known through noisy observational increments. In the real applications, we compute the low-order subspace of the bred vectors from the analysis, not the true state, but experience indicates that the bred vectors are insensitive to typical errors in the analysis (Corazza et al. 2002a,b).

a. Growth of initial forecast errors

Because developing a data assimilation system for the ZC model is not the goal of this study, we illustrate the potential benefits of using bred vectors in data assimilation through the use of six sets of special forecast experiments. Each set consists of 12-month forecasts started at the beginning of each calendar month of the control run for a period of 500 years, from year 11 to 510, giving a total of 6000 cases. The initial state of the forecast experiments is perturbed by adding six different types of assumed initial errors. The size of the initial errors is calculated according to

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where X stands for the initial error fields and the subscript i represents one of the six variables: oceanic currents, thermocline depth, SST, and atmospheric winds. The values of σ_i (i = 1, 2, … , 6) are calculated according to (2). The size of the initial errors in all forecasts is fixed to be /10.0, which corresponds to 10% of the control run’s variability. Table 3 summarizes the six types of initial errors used in the forecast experiments.

Note that in the BV (or −BV) experiment the initial error is a function of the starting time, but in the CBV experiment the initial error is only a function of the background ENSO phase at the starting time, as defined in the 24 bins listed in Table 2. In other words, each initial state for the BV (or −BV) experiment has a different initial error field (the bred vectors obtained for that month), but there are only 24 different initial error fields for the CBV experiment.

To construct initial error fields for the NO_BV experiments, we remove the projection using

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where R_i stands for one of the six variables in the NO_BV initial error fields, and F_i and B_i correspond to the STND and BV initial error fields, respectively. Note for the sake of generality, we used different symbols to denote the size of initial errors in BV and STND (σ_B, σ_F, respectively) in (8) although σ_B = σ_F in our experiments. The L2 norm of the STND and BV initial errors are σ_F and σ_B, respectively, and γ is the inner product of the STND and BV initial errors, which is calculated according to

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As in (1)–(3), the integrations in (7)–(9) are carried out over the ZC model domain. It is easy to verify that the L2 norm of the NO_BV initial errors constructed with (8)–(9) is the same as that of the STND experiments (i.e., σ_R = σ_B = σ_F). Similarly, we can use (8)–(9) to obtain the initial error fields for the NO_CBV experiments. We have confirmed that the results of NO_BV and NO_CBV experiments are similar when the ocean energy norm is used instead of the L2 norm.

The forecasts are verified against the control run using a “perfect model” scenario. The forecast errors are measured by the same L2 norm defined in (7) except X now stands for the difference between the forecast and control runs. The results of the forecast experiments are summarized in the form of the averaged forecast errors as a function of the forecast lead time and the month in which the forecasts are verified (referred to as the “target month”).

Figure 7 displays the results for experiments 1–3 (BV, −BV, and CBV). The most striking feature of this figure is the relatively large forecast error associated with forecasts started around spring and verifying at short lead times in the summer, then with longer lead times in the fall and winter. Note that the largest forecast errors with longer lead times are associated with forecasts initiated in successively earlier months in the year. The higher loss of skill for forecasts started in the spring is a well-known feature of ENSO forecast errors and has been referred to in the literature as the “spring barrier.” It has been observed in simple anomaly coupled models (e.g., Cane 1991), in fully coupled OGCM systems (e.g., Latif et al. 1998), and in statistical forecast models (H. Van den Dool 1999, personal communication). Figure 7 shows that the spring barrier is strongly present when the initial “errors” are pure BV or CBV. It can also be seen that the error growth of −BV is very comparable with that of BV, particularly when the lead time is less than 6 months.

The spring barrier is even more pronounced when the composite bred vectors CBV are used as perturbation, due to their higher growth rates, especially during the critical spring months. A single BV, presumably due to nonlinear effects, contains some noise that is filtered out by compositing all the BVs over 1000 years according to the phase of the ENSO. Apparently such a composite provides a better estimate of the fastest-growing vector in a nonlinear environment than a single BV. Although a composite over many cases such as we have performed here is not feasible with a fully coupled model, an EOF expansion of the bred vectors may capture the advantages of the composite bred vector.

Figure 8 is the counterpart of Fig. 7 with forecast errors plotted for experiments 4–6 (STND, No-BV, and No-CBV). Note that the error growth for the STND experiment (Fig. 8a) is smaller than that for the BV runs (Figs. 7a–c) all year round, indicating, as expected, that random perturbations exhibit less growth than the BVs. For example, while random initial errors in forecasts verifying in June double in approximately 3 months, CBV errors double in less than half of that time period. Yet the spring barrier still dominates the forecasts with STND initial errors. Eliminating from the random initial errors their component projecting on BVs and especially on CBVs (while maintaining their size), however, substantially mitigates the effect of the spring barrier. For example, the error doubling time for forecasts verifying in June is extended from 3 to close to 5–6 months when no BV/CBV error is permitted in the initial STND errors, with a maximum rms error reduction of 15%–30%.

As discussed in the introduction, data assimilation cycles behave in some sense like breeding cycles and accumulate flow-dependent errors along the leading LVs or BVs. This has been conjectured by Toth and Kalnay (1993) and subsequently confirmed by a number of studies including that of Pires et al. (1996), Swanson et al. (1998), Smith and Gilmour (1998), COR, and Corazza et al. (2002b), see also related discussion in Toth et al. (1999). The skill of data assimilation systems could therefore be potentially improved by either removing the projection of bred vectors from the observational increments (observations minus first guess) or by using the bred vectors to estimate the background error covariance matrix (see, e.g., Kalnay and Toth 1994; Corazza et al. 2002a,b). In both approaches, the bred vectors define a low-dimensional subspace with few degrees of freedom, and the availability of many observations allows their removal from the observational increments even without knowing the true state of the coupled system.

The results presented in Figs. 7–8 illustrate the potential beneficial impacts by removing the projection of bred vectors from the observational increments (observations minus first guess) in an operational forecast system. Specifically, the difference between the STND (Fig. 8a) and NO_BV (Fig. 8b) experiments can be viewed as a lower bound for the expected benefit from the use of a single breeding cycle in data assimilation in a perfect model scenario. The comparison of Figs. 7a (BV experiment) and 8b (NO-BV), on the other hand, can give an estimate on the upper limit of the benefit of using the bred vectors in ZC model in terms of error reduction within a single data assimilation cycle. This upper limit of error reduction is about 25%–30%. In a real operational forecast environment, the error reduction is expected to be smaller due to imperfections of the model and the uncertainty in estimating errors as well as bred vector patterns. In addition, the potential beneficial impact of removing the projection of bred vectors from the observational increments may depend on the relative importance of non–BV component versus BV component in the initial errors fields.

6. Conclusions

In this study, we applied the breeding method to the Zebiak–Cane coupled atmosphere–ocean model (ZC model) to demonstrate the feasibility of using the breeding method in the context of data assimilation and ensemble forecasting for ENSO events in a perfect model scenario. Since the ZC model has no weather or other fast timescale instabilities it offers the simplest possible setting to test the potential utility of the breeding method for ENSO predictions. With the ZC model, we have little ambiguity in relating the fast-growing modes identified with the breeding method to the background flow (including both the seasonal cycle and the ENSO events). As a result, we could make a series of experiments to illustrate the potential benefit of using BVs in reducing the forecast errors for ENSO prediction.

We found that both the spatial structure and the growth rate of bred vectors of the ZC model are strongly related to the background ENSO events. The spatial structure of the bred vectors tends to be more coherent and with a spatial structure reminiscent of the ENSO anomaly itself (see plots of composite BV, Fig. 4). At the extreme phases (either the cold or warm event), the spatial structure of bred vectors has much less projection on the ENSO mode. For a given background ENSO, the bred vectors tend to be out of phase, with maximum amplitude during the transition between cold and warm events, and can have either a positive or a negative Niño-3 index. Often the bred vectors tend to alternate from one sign to the opposite sign when the background ENSO cycle passes through the extreme phase. In accordance with the spatial structure, the bred vectors grow more vigorously several months prior to and after an El Niño event. At the mature stage of an El Niño event, the growth rate is nearly neutral. A simple delayed oscillator model explains qualitatively the dependence of the bred vector growth rate on the background ENSO cycle (P. S. Schopf 2000, personal communication). The growth rate of the bred vectors in the ZC model is also strongly dependent on the seasonal cycle, growing most vigorously during the summer season and becoming nearly neutral in the late fall or early winter. This is in agreement with earlier studies with the ZC and Battisti models using singular vectors, when the singular vectors are allowed to evolve over an optimization period longer than 3 months, in which case the evolved leading singular vectors approach the bred vector (e.g., Chen et al. 1997; Xue et al. 1997a,b; and Thompson and Battisti 2000, 2001).

We have explored the applications of the bred vectors in two contexts: data assimilation and ensemble forecasting. Several “perfect model” forecast experiments have been made to illustrate the potential benefits of using bred vectors for ENSO predictions, using pairs of added and subtracted bred vectors.

We have shown that the error amplification is much stronger when the initial errors are made of bred vectors (referred to as the BV experiments) than random initial errors (referred to as the STND experiments). This is particularly true for the forecasts started in the spring season. The larger forecast error observed for the forecasts starting from the spring season is known as the spring barrier. The much stronger seasonal dependence of the forecast errors in the BV experiments suggests that part of the spring barrier in the ZC model is associated with the presence of the bred vectors in the initial error field. This has been confirmed by experiments where the bred vectors have been removed from the initial error field (referred to as the NO_BV or NO_CBV experiments), which show a systematic reduction of the forecast error over all seasons, and in particular result in a less noticeable spring barrier.

Our results indicate that if the projection of random initial errors on BVs is eliminated (while keeping the size of the error the same), the forecast errors are reduced systematically (up to 30%) while the doubling time of forecast errors can be delayed by a month or more. Toth and Kalnay (1993) conjectured that the errors in a first guess field of a data assimilation system have a strong projection on the dynamically growing directions that can be captured by BVs. In this case removing these dynamically growing errors inherited in the first guess fields would yield an even larger reduction of forecast errors.

Three sets of ensemble forecast experiments have been made to assess the beneficial impact of using bred vectors as the choice of the ensemble perturbations for ENSO prediction with the ZC model in the context of perfect model scenario. In these ensemble forecast experiments we found that the ensemble mean forecasts are always more skillful than their control forecasts. When dynamically growing errors are present in the initial error fields, the ensemble forecasts with a pair of composite bred vectors yield a significant improvement compared to those with a pair of random perturbations. Moreover, the reduction of the forecast errors with a pair of bred vectors is stronger when the errors in the control forecast experiments, associated with the spring barrier, grow more vigorously. As a result, the spring barrier exhibited in the control forecast is more reduced in the ensemble forecasts using the bred vectors. However, when the initial control forecast errors contain no error along the BVs, the ensemble forecasts with a pair of bred vectors exhibit little advantage over those with a pair of random maps. In these cases, the error reduction is fairly uniform over all seasons for both types of ensemble perturbations.

Although these experiments have been done assuming perfect knowledge of the true state of the coupled system, and with BV bred using the true state, the results should be applicable to a more realistic situation in which we only have noisy observations, and the bred vectors are computed from the analysis, not the truth. This is because the bred vectors are not sensitive to the details of the flow, and noisy observational increments are enough to estimate the projection of the forecast error on the subspace of the bred vectors.

This study has laid the foundation for the next phase of our study where we plan to apply the breeding method in a fully coupled atmosphere–ocean GCM model system to improve coupled model data assimilation and ensemble forecasting for seasonal climate prediction. When dealing with a more realistic CGCM system, we need to isolate in the BVs the ENSO-related variability and its associated global atmospheric–oceanic variability from that caused primarily by uncoupled processes, such as weather or other high/low-frequency atmospheric–oceanic variability with nearly equal amplitude. Experiments performed with a comprehensive CGCM will show whether the breeding technique, which is computationally efficient, will enable us to separate the ENSO signal from other modes, and the extent to which it contributes to reduce forecast errors, either through ensemble forecasting or by providing information about the evolving error covariance.