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    Real (dash line) and imaginary (solid line) parts of the largest eigenvalue (sorted by real part) of the linear operator 𝗔 as a function of coupling parameter β. The system becomes unstable at approximately β = 1/112.25 (day−1), at which point the real part of the eigenvalue changes from negative values to positive values. For coupling parameter β between 1/900 and 1/112 (day−1), the dominant eigenmode shows a damped oscillation with periods greater than 6 yr

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    Time–latitude plots of the dominant eigenmode whose eigenvalue is shown in Fig. 1 for coupling parameter β of (a) 1/120, (b) 1/200, and (c) 1/500 (day−1). For illustration purposes, the exponentially decaying part of the solution is omitted. The contour interval is 0.05. The oscillation period is given by the imaginary part of the eigenvalue in Fig. 1. Note that the modal structure changes from a “dipole” to a “monopole” as the coupling parameter decreases

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    Power spectrum of (a) the coupled system and (b) the corresponding normal system as a function of coupling parameter β (see text for their definitions). The dash–dot lines indicates the period of the dominant eigenmode shown as the imaginary part in Fig. 1. The dashed line denotes the maximum spectral power density or spectral “peaks” at a given coupling parameter. Note that energy level in the full system is substantially higher than that of the normal portion of the system. A nonuniform contour interval (approx logarithmic) is used. Shading indicates that the power densities are greater than a nondimensional value of 100. The spectra for selected values of the coupling parameter, (c) β = 1/500, (d) 1/200, and (e) 1/120 day−1. In each the solid line represents the spectrum of the nonnormal system, the dashed line represents the spectrum of the normal system, and the dash–dot line represents the spectrum of the degenerate system where both V(y) and β are zero

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    Pseudospectra of the coupled system for coupling parameter β = 1/200 day−1. The points indicate the first 30 eigenvalues of the system. The contours indicate pseudospectra for different values of ϵ ranging from 10−12 to 10−7

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    Normalized error variance growth of the full system, as expected from the theory under a unitary forcing as a function of coupling parameter for (a) the full system and (b) its normal counterpart. (c) The difference between the two error variances. The error variance of each system is normalized by its own total variance. A contour interval of 0.1 is used in (a) and (b). The contour interval in (c) is 0.05. The thick contour line of value 0.5 gives the lead times when the system's signal-to-noise ratio reaches unity. Error variance of the (d) nonnormal and (e) normal and (f) the difference between the two systems for selected values of coupling parameter, β = 1/120 (solid), 1/200 (dash), and 1/500 (dash–dot) day−1. In each the dotted line shows the error variance of the degenerate system where both V(y) and β are zero

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    Theoretical persistence skill of the coupled system under a unitary forcing as a function of coupling parameter β for (a) the full system and (b) its normal portion. (c) The difference between the two correlation skills. A contour interval 0.1 is used in (a) and (b). The contour interval in (c) is 0.05. Persistence of the (d) nonnormal and (e) normal and (f) the difference between the two systems for selected values of coupling parameter, β = 1/120 (solid), 1/200 (dash), and 1/500 (dash–dot) day−1. In each the dotted line shows the persistence of the degenerate system where both V(y) and β are zero

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    Theoretical correlation skill of the coupled system under a unitary forcing as a function of coupling parameter β for (a) the full system and (b) its normal portion. (c) The difference between the two correlation skills. A contour interval 0.1 is used in (a) and (b). The contour interval in (c) is 0.05. Persistence of the (d) nonnormal and (e) normal and (f) the difference between the two systems for selected values of coupling parameter, β = 1/120 (solid), 1/200 (dash), and 1/500 (dash–dot) day−1. In each the dotted line shows the persistence of the degenerate system where both V(y) and β are zero

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    (a) Optimal noise forcing structure as a function of lead time τ. The pattern has been normalized to unit variance at each lead time. The horizontal line marks the lead time at which the signal-to-noise ratio is equal to unity, i.e., ϵ2(τ) = χ2(τ) = 0.5. (b) Stochastic optimal (dashed) and the optimal noise forcing fϵ2=0.5 (solid). (c) Error variance ϵ2(τ) growth under stochastic optimal (dashed), the optimal noise forcing fϵ2=0.5 (thin solid), and the optimal noise forcing which maximizes the predictictability at each lead time (thick solid)

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    (a) The first predictable pattern as a function of lead time τ. The pattern has been normalized to unit variance at each lead time. The horizontal line marks the lead time at which the signal-to-noise ratio is equal to unity, i.e., ϵ2(τ) = χ2(τ) = 0.5. (b) The EOF that corresponds to the first predictable pattern at lead time τ = 0 (dashed) and the first predictable pattern at lead time τ of unity for the signal-to-noise ratio (solid). (c) Error variance ϵ2(τ) growth associated with the EOF (dashed), the first predictable pattern of unity signal-to-noise ratio (thin solid), and the patterns that minimize the error growth at each lead time (thick solid)

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    Numerical estimates of the model's predictive skills using a Monte Carlo simulation with a verification period of (a) 10, (b) 20, (c) 50, and (d) 100 yr. Each set of experiments consists of 1000 simulations. The dashed lines represent the ensemble averages of predictive skills and the shading indicates standard deviations within the ensemble. The solid lines represent the theoretically expected predictive skills. It is evident that the numerical estimates converge to the theoretical solutions as verification period increases. The numerical experiments were conducted using a uniform noise forcing with coupling parameter β = 1/200 (day−1)

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Predictability of Linear Coupled Systems. Part II: An Application to a Simple Model of Tropical Atlantic Variability

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  • 1 Department of Oceanography, Texas A&M University, College Station, Texas
  • | 2 National Center for Atmospheric Research, Boulder, Colorado
  • | 3 Department of Oceanography, Texas A&M University, College Station, Texas
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Abstract

A predictability analysis developed within a general framework of linear stochastic dynamics in a companion paper is applied to a simple coupled climate model of tropical Atlantic variability (TAV). The simple model extends the univariate stochastic climate model of Hasselmann by including positive air–sea feedback and heat advection by mean ocean currents. The interplay between the positive air–sea feedback and the negative oceanic feedbacks gives rise to oscillatory coupled modes. The relationship between these coupled modes and the system's predictability is explored for a wide range of coupled regimes. It is shown that the system's predictability cannot be simply determined from oscillatory behavior of the dominant coupled mode when coupling is weak. The predictable dynamics of the weakly coupled system depend upon the interference among many coupled modes and the spatial structures of the stochastic forcing. Using the simple model as an example, the concept of optimizing the predictability of a linear stochastic system is illustrated. The implication of these results for the predictability of weakly coupled climate systems in the real world is also discussed.

Corresponding author address: Dr. Ping Chang, Department of Oceanography, Texas A&M University, College Station, TX 77843-3146. Email: ping@ocean.tamu.edu

Abstract

A predictability analysis developed within a general framework of linear stochastic dynamics in a companion paper is applied to a simple coupled climate model of tropical Atlantic variability (TAV). The simple model extends the univariate stochastic climate model of Hasselmann by including positive air–sea feedback and heat advection by mean ocean currents. The interplay between the positive air–sea feedback and the negative oceanic feedbacks gives rise to oscillatory coupled modes. The relationship between these coupled modes and the system's predictability is explored for a wide range of coupled regimes. It is shown that the system's predictability cannot be simply determined from oscillatory behavior of the dominant coupled mode when coupling is weak. The predictable dynamics of the weakly coupled system depend upon the interference among many coupled modes and the spatial structures of the stochastic forcing. Using the simple model as an example, the concept of optimizing the predictability of a linear stochastic system is illustrated. The implication of these results for the predictability of weakly coupled climate systems in the real world is also discussed.

Corresponding author address: Dr. Ping Chang, Department of Oceanography, Texas A&M University, College Station, TX 77843-3146. Email: ping@ocean.tamu.edu

1. Introduction

Predictability in the climate system is often associated with coupled interactions between its different components. The component whose predictability is of the most interest to humans is the atmosphere. However, in the absence of boundary forcing, the atmosphere tends to lose its memory of initial condition in less than a month. For prediction on seasonal and longer time scales, knowledge of the boundary conditions that force the atmosphere becomes very important. Of particular importance is the role of the ocean surface: in contrast to the atmosphere, the ocean retains its memory of initial condition for a much longer period, ranging from seasonal to interannual time scales over much of the ocean surface, to millennial time scales at depth.

The simplest example of air–sea coupling enhancing the predictability of the atmosphere is provided by considering an energy balance coupled model (Barsugli and Battisti 1998). In this modeling framework, dynamics of the atmosphere and ocean do not come into play, the enhanced predictability of the atmosphere simply comes from the vast difference in thermal inertia between the atmosphere and the ocean. The much larger heat capacity of the ocean gives a longer damping time scale of the temperature anomalies in the ocean mixed layer, which in turn feeds back to the atmosphere via “passive coupling,” hence, enhancing the memory of the atmospheric anomalies (Bretherton and Battisti 2000). In Part I of this study (Chang et al. 2004), we demonstrated, using a very simple prototype model, how this differential damping mechanism works when the atmosphere and ocean are passively coupled. However, the difference in the damping time scales between the atmosphere and the ocean is just one of the important features of the coupled system. Other important features include nonlocal feedbacks associated with atmospheric and oceanic advection and wave adjustment processes. These processes become important when there is an “active coupling” between the atmosphere and ocean, which usually involves a positive air–sea feedback. How the active coupling contributes to the predictability of the coupled system depends on the nature and strength of the feedbacks.

One of the best known examples of coupled phenomena where active coupling plays a vital role is El Niño–Southern Oscillation (ENSO) in the tropical Pacific. The key positive air–sea feedback in this case is the Bjerknes feedback (Bjerknes 1969), which operates between the trade winds and equatorial sea surface temperature. This feedback is hypothesized to be sufficiently strong so that the coupled system may be unstable and the evolution of ENSO may be characterized by the most unstable coupled mode [see Neelin et al. (1998) for a review]. In this one-mode-dominant system, the predictable dynamics largely depend upon the characteristics of the leading coupled mode.

However, not all coupled phenomena in the climate system involve strong positive air–sea feedbacks. In fact, one could probably argue, on the basis of available observational and modeling evidence, that most coupled phenomena, with the exception of ENSO, involve only weak positive air–sea feedbacks. The weak feedbacks are unlikely to cause coupled instability, and thus the coupled response in general cannot be characterized by a single dominant coupled mode. We loosely refer to those coupled systems, where the positive air–sea feedbacks play some role, but are not strong enough to allow one coupled mode to dominate, as weakly coupled systems. The purpose of this study is to explore the predictable dynamics of a weakly coupled system. Here we would like to apply the theoretical tools derived in Part I to a more physically motivated model of the ocean–atmosphere system, one which incorporates both the differential damping effect and nonlocal feedback processes.

We choose a simple coupled model of tropical Atlantic variability (TAV) as a case study. It can be argued that the TAV coupled system falls in the category of weakly coupled systems. Several recent modeling studies (Zebiak 1993; Chang et al. 1997, 2001) have suggested that positive air–sea feedbacks operate in the deep Tropics of the Atlantic. While Zebiak (1993) investigated a dynamical feedback akin to ENSO in the tropical Atlantic, Chang et al. (1997) focused on the positive thermodynamic feedback between wind-induced heat flux and sea surface temperature anomalies. These studies converge on a similar notion that self-sustaining coupled oscillations are unlikely to be present in the tropical Atlantic region. This led Chang et al. (2001) to propose that TAV may be best described as a stable dynamical system driven by stochastic processes. Its linear stochastic dynamic framework makes TAV an ideal example to illustrate the complexity of the predictable dynamics of weakly coupled climate systems. Therefore, in this study we shall use the predictability analysis tools developed in Part I to dissect the predictable dynamics of a coupled TAV model.

Apart from the reason that the TAV system is believed to be weakly coupled and obeys stable linear stochastic dynamics, we choose this system for the case study for two other considerations. First, the tropical Atlantic is perhaps the region that holds the most promise for providing climate predictability that is not directly related to ENSO. Yet, the predictable dynamics of TAV are largely unexplored at present, except that the remote influence due to ENSO has been shown to add some predictability to the SST variability in the northern tropical Atlantic (Penland and Matrosova 1998). It is, thus, of considerable interest to gain fundamental understanding of how local feedbacks can potentially enhance the predictability of TAV. Second, Chang et al. (2001) presented a simplified coupled model of TAV that can be conveniently cast into an autonomous multivariate stochastic system of the following form:
i1520-0442-17-7-1487-e1
The properties of the system can be studied elegantly using the tools developed in Part I. The simplicity of the model formulation permits a detailed and systematic exploration of important dynamic factors determining the predictability of the system. Yet, the physical processes included are realistic enough to shed light on important issues concerning the predictability of the real climate system. Therefore, the results presented here should have implications for a weakly coupled climate system in general.

In the context of TAV, there have been suggestions of a weak spectral peak at subdecadal time scales in the interhemispheric SST anomaly of the tropical Atlantic (Mehta 1998). The existence of the weak spectral peak seems to be consistent with the notion that the TAV system is weakly coupled and supports a damped oscillatory coupled mode, although the issue of a true spectral peak remains controversial. Rather than participating in the debate about the weak spectral peak, we assume that such a peak exists. Our investigation then focuses on the issue of whether the weak peak contributes to an enhancement of the predictability of the coupled TAV system.

The study is organized as follows. In section 2, a simple coupled model of TAV is introduced. The model extends the univariate stochastic climate model for SST (Hasselmann 1976) by including a positive air–sea feedback and heat advection by ocean mean currents. In section 3, dynamical properties of the governing system are explored for a wide range of coupled regimes. In section 4, predictability and predictable dynamics of the system are analyzed using the tools developed in Part I. In section 5, numerical simulations are carried out to verify the theoretical results and to address sampling related issues. Finally, in section 6, we summarize and discuss the major findings and implications for the tropical Atlantic region and the weakly coupled climate system in general.

2. A simple coupled model of TAV

Chang et al. (2001) presented a simple coupled model that focused on the interhemispheric SST variability in the tropical Atlantic. Derivation of the simple model was based on results of uncoupled and coupled GCM analysis, as well as empirical studies of observations. The simplification was achieved by first considering a zonally averaged temperature equation. This was justified by the observation that the spatial pattern of SST anomalies in the tropical Atlantic consists primarily of banded structures with little zonal variation. Then, it assumes that a positive air–sea feedback takes place in the tropical Atlantic ITCZ region. This assumption was supported by recent atmospheric GCM studies by Chang et al. (2000) and Saravanan and Chang (2000), which show that a positive correlation relationship exists between wind-induced latent heat flux and SST in the deep Tropics of the Atlantic sector. Finally, the change of upper-ocean heat transport is assumed to be regulated through the advection of anomalous temperatures by the mean meridional current and equatorial upwelling. This approximation is based upon the results of the hybrid coupled GCM study of Chang et al. (2001) and the ocean GCM study of Seager et al. (2001), which show that the advection of anomalous temperatures by the mean current dominates the changes in surface ocean heat transport in the tropical Atlantic Ocean. Taking all the simplifications into the consideration, one arrives at a simple coupled model for the zonally averaged SST anomaly:
i1520-0442-17-7-1487-e2
where V(y) is the zonal-averaged mean meridional velocity; D(y) is a Newtonian damping term resulted from the mean upwelling; κ is a diffusion coefficient, and Q is the anomalous surface heat flux that consists of an air–sea feedback component Qc and a stochastic component Qs. The former can be presented in a simple manner Qc = βS(y)T(t), where β is a coupling parameter controlling the strength of air–sea feedback, T(t) is the SST anomalies averaged within the tropical Atlantic ITCZ region where the atmosphere is sensitive to SST changes, S(y) is a spatial structure function. The latter can be expressed as Qs = Σi fi(y)ηi(t), where fi(y) represents a spatial distribution of the ith component of the random noise forcing and ηi(t) is a Gaussian white noise process, representing weather effects at all frequencies. Chang et al. (2001) show that the governing simple model (2) captures major features of the interhemispheric SST variability in the HCM. A detailed discussion on the relevance of the simple model to TAV can be found in Chang et al. (2001).

Following Chang et al. (2001), the meridional mean velocity V(y) and the Newtonian damping D(y) were taken from their hybrid coupled model. The T(t) is averaged between 8° and 12°N. The spatial structure of the surface heat flux S(y) is determined by regressing the net surface heat flux anomaly of the Comprehensive Ocean–Atmosphere Data Set (COADS) onto T(t) derived from an observed SST anomaly [see Chang et al. (2001) for details]. The diffusion coefficient κ was assigned a value of 1 × 108 cm s−1. This value was used throughout the analysis, because the analysis was not particularly sensitive to the choice of the diffusion coefficient. A no-flux boundary condition was used to ensure that the SST anomalies are generated locally within the deep Tropics.

The differential equation (2) can be discretized in space by employing a centered finite difference in y. The convergence of the discrete solution to its exact solution is guaranteed by the Lax equivalent theorem (Fletcher 1988). The spatial discretization leads to a multivariate linear stochastic system in form (1) with θ being a state vector of the SST anomaly that contains SST anomalies at all grid points. The 𝗔 includes dynamic processes, such as the positive air–sea feedback between surface heat flux and SST, advection of heat by mean currents, and diffusive effect. The results presented below were based on a grid resolution of 0.5° within 30°S–30°N, which gives a dimension of 121 for the state vector θ and a matrix dimension of 121 × 121 for 𝗔. The findings of this study are not sensitive to grid resolution.

As indicated earlier, the response and predictability of the governing linear stochastic system depend on the dynamic operator 𝗔 and forcing matrix 𝗙. In the absence of air–sea coupling, meridional heat advection and diffusion processes, that is, β = 0, V(y) = 0, and κ = 0, and 𝗔 reduces to a diagonal matrix, so that the temporal evolution at each location becomes independent of other locations and is determined by the local damping rate. In this case, (1) degenerates into a set of uncoupled univariate stochastic climate models of the type discussed by Hasselmann (1976). In the following sections, we shall dissect this simple coupled system to examine how these additional dynamical processes can affect the predictability.

3. Dynamics of the simple TAV model

As shown in Part I, the predictability of a linear stochastic system (1) depends on the dynamics of the deterministic operator 𝗔 and stochastic forcing 𝗙. The dynamics of 𝗔 is affected by the coupled dynamics, which in turn can affect the temporal and spatial characteristics of the eigenmodes, thereby having an impact on the predictability of the coupled system. Therefore, before embarking on the investigation of the predictability, it is important to first take a look at how the coupled modes behave as the strength of the air–sea coupling varies.

a. Eigenmodes

The coupled modes are given by the eigenmodes of 𝗔. Of particular interest is the least damped mode for a stable system or the fastest growing mode for an unstable system, because it characterizes the free system's response to an initial perturbation as time approaches infinity. Figure 1 shows the eigenvalue of the least damped eigenmode as a function of coupling parameter β. The solid and dashed lines represent the real and imaginary part of the eigenvalues. The real part of the eigenvalue becomes greater than zero when the coupling parameter β is larger than 1/112 (days−1), indicating that the system becomes unstable beyond this point. This result is consistent with the numerical solutions of Chang et al. (2001). For coupling parameter β varying between approximately 1/900 to 1/112 (days−1), the system has the least damped oscillatory eigenmode with a period varying from subdecadal time scales (6 years) to multidecadal time scales. Chang et al. (2001) show that this oscillatory mode results from an interplay between the positive air–sea feedback and the negative feedback due to the advective heat transport of the ocean. As expected, the damping rate of the mode increases as the coupling strength decreases. When the coupling is sufficiently weak, no oscillatory mode exists and all the eigenmodes decay exponentially with time.

The spatial structure of the leading eigenmode is displayed in Fig. 2 as a time–latitude plot. For illustration purposes, the exponentially decaying part of the solution is excluded in all the plots so that spatial structures of the modes can be clearly seen. It is interesting to note how the structure of the dominant mode changes as coupling strength varies: in the strong coupling regime, the dominant mode has an appearance of a “dipole,” that is, variations in the two hemispheres tend to be out of phase. As the coupling strength weakens, this dipolelike structure gradually evolves into a monopole-like structure with large amplitude in the coupling hemisphere, and variations in both hemispheres become nearly in phase. This suggests that a dipolelike mode may only exist when the air–sea coupling is strong, which is also consistent with the hybrid coupled model study of Chang et al. (2001). They further argue that the estimated coupling strength in reality is probably close to β = 1/200 (days−1). This puts the coupled system in a moderately coupled regime, well below the critical value for a self-sustained oscillation.

The fact that the air–sea feedback strength does not only affect the temporal characteristics, but also the spatial characteristics of the eigenmodes indicates that the coupling between the atmosphere and oceans can change the degree of linear interference among the coupled modes (or eigenmodes). In other words, the coupling can alter not only the damping rate and oscillatory behavior, but also the nonnormal growth of the system.

b. Power spectrum

Next we consider the temporal power spectrum of variability in the coupled system. For time t sufficiently larger than the initial time t0, that is, tt0, or a large lead time τ = tt0 ≫ 0, the effect of initial condition dies away and the solution to (1) is given by the unpredictable component that describes the climatological response of the system:
i1520-0442-17-7-1487-e3
Let ω be the frequency of the response and θ̂(ω) be the Fourier transform of the forced response θ. It can be readily shown that the power spectrum of the response integrated over the spatial domain is given by
i1520-0442-17-7-1487-e4
where 𝗥(ω) = (𝗜 − 𝗔)−1 is called the resolvent of 𝗔. It is evident from (4) if 𝗙𝗙* = 𝗜, that is, noise forcing is unitary, then the spectrum E(ω) is completely determined by the dynamic operator 𝗔. Farrell and Ioannou (1996) show that for a normal matrix 𝗔, the power spectrum E(ω) is given by E(ω) = Σi |λi|−2, where λi are eigenvalues of 𝗔. This means the spectrum of a normal system only depends on the eigenvalues of the eigenmodes (normal modes). In this case, if the system supports oscillatory normal modes, resonant responses will occur near the frequencies of these modes. Close to these frequencies, one expects to find spectral peaks. In particular, if the response is dominated by a single oscillating mode, a pronounced spectral peak should be easily identified.

For a general system where the eigenmodes are not orthogonal, the spectrum of the system's response can be altered by the interference among the eigenvectors. As we know, this interference always acts to enhance the climatological variance of the system provided that the noise forcing is unitary. This means that the overall spectral power of a nonnormal system is higher than that of the corresponding normal system. Ioannou (1995) shows that under a unitary noise forcing, nonnormality enhances the spectral power at all frequencies. Furthermore, it is readily shown that in the high-frequency limit (ω → ∞), the power spectrum E(ω) → Σi ω−2, which is independent of the dynamics of the system. This result implies that the spectrum of a nonnormal system may be generally redder than the normal counterpart. Therefore, identifying a significant low-frequency spectral peak may be more difficult in a nonnormal system than in a normal system.

To illustrate this finding, we compare the power spectrum of the simple coupled system, which is nonnormal, to the spectrum of the corresponding normal system where we effectively alter the spatial structure of the coupled modes so that they become orthogonal to each other, while keeping the eigenspectrum intact. As expected from the theory, Fig. 3 shows that the overall spectral level of the nonnormal system is substantially higher than that of the normal system. The spectral power enhancement is particularly significant in the low-frequency range, making the spectrum of the nonnormal system generally redder than that of the corresponding normal system. As a result, the spectral peak of the normal system appears to be more pronounced than that of the actual coupled system. For both systems, the maximum spectral peaks tend to occur at lower frequencies than the frequencies of the leading eigenmode. This shift toward the lower frequency appears to be more evident in the nonnormal system than in its normal counterpart.

Pseudospectrum analysis (e.g., Trefethen 1997) gives a more intuitive understanding between nonnormality and response power spectrum. Recall that pseudospectra is defined as the set of ω at distance ≤ϵ such that the norm of the resolvent ‖𝗥(ω)‖ is sufficiently large (≥ϵ−1). For a unitary noise, the response power spectrum E(ω) depends solely on the resolvent. Therefore, it is directly linked to pseudospectra of 𝗔. Figure 4 shows the pseudospectra of the simple TAV model for coupling parameter β = 1/200 day−1, computed using the Matlab program provided by Trefethen (1999). It is evident that the areas surrounded by the contours of constant ϵ are irregular in shapes and are much larger than those expected from a normal system, which would be the areas of disks of radius ϵ centered at each individual eigenvalue. Physically, this means that the resonance can occur even when forcing frequency is far from the eigenspectrum. Therefore, the overall spectral level of the system is substantially higher than that of the normal system.

Although the above comparison between the nonnormal system and the corresponding normal system is meaningful in a theoretical sense, it is often difficult in practice to physically keep the system's eigenspectrum intact while allowing the spatial structure of the eigenmodes to change. This is because a certain physical process in the dynamical operator 𝗔 affects both the temporal and spatial characteristics of the eigenmodes. Therefore, when the nonnormal growth of the system is removed by eliminating certain physical processes, the temporal characteristics of the eigenmodes are also altered.

To illustrate this point, in the bottom panels of Fig. 3 we compare, for three selected coupling parameter values, the spectra of the nonnormal model and the corresponding normal system with the spectrum of a degenerate system where we set both the mean meridional current V(y) and the coupling parameter β to zero. The degenerate system essentially reduces to the Hasselmann model (except for the diffusion operator which does not change the basic properties of the system). From earlier discussions, we know that this system is trivially normal and contains no oscillatory modes. The spectrum is, therefore, red, as indicated by the dot-dash lines in Figs. 3c,d,e. When the coupling is strong [e.g., β = 1/120 (days−1)], the normal system shows a resonant response with most of its energy concentrated near the frequency of the dominant normal mode, whereas the spectrum of the nonnormal coupled model has a significantly enhanced background power in addition to the peak near the resonant frequency. The spectrum of the degenerate model seems to fit nicely with the background spectrum of the normal system, which is considerably lower than that of the nonnormal system.

Therefore, in the strongly coupled regime the air–sea feedback and ocean heat transport not only give rise to an oscillatory behavior in the coupled system, but also contribute to a substantial increase in the variance due to enhancing nonnormality of the system. When the coupling strength decreases, the spectral peaks are less well defined in both the nonnormal and the normal systems. The spectrum of the degenerate system generally has higher power than that of the normal system and resembles more closely the nonnormal coupled system spectrum, particularly in the low-frequency range. Because nonnormal behavior of the system is expected to influence the power spectrum in the low-frequency range, this result suggests that, at least in this simple coupled system, air–sea feedback is the prime contributor to the nonnormal behavior. This result is consistent with a direct inspection of the structure of the dynamic operator 𝗔. It shows that the asymmetry of 𝗔 comes mainly from the nonlocal effect in the air–sea feedback term, Qc = βS(y)T(t).

It is worth noting that the spectrum of the nonnormal model bears the closest resemblance to the observed spectrum of the tropical Atlantic SST dipole index (see Fig. 15 of Chang et al. 2001) when the system retains a moderate coupling strength, β = 1/200 (days−1). This suggests again that a moderate coupled regime may be most relevant to TAV in reality.

4. Predictability analysis

Having examined the dynamical properties of the simple coupled model, we now turn to the predictability analysis of the simple coupled model using the error variance–based measures introduced in Part I. Following the approach outlined in Part I, we first individually investigate the effects of deterministic dynamics in operator 𝗔 and stochastic forcing 𝗙 on predictability. We then discuss the maximum predictability for a given dynamical operator 𝗔 and stochastic forcing 𝗙.

a. Effect of coupled dynamics

We first compare the predictability of the nonnormal system to the corresponding normal system, in order to gain theoretical insight into the effect of the dynamical operator 𝗔 on predictability. Because the normal and nonnormal systems have the same eigenspectrum but differ in eigenvector structures, this comparison gives a quantitative estimate of the contribution from eigenmode interference to the predictability.

Figures 5a,b show the normalized error variance of both systems under the same unitary noise forcing. The normalized error variance is defined as (see section 3c in Part I for discussion),
i1520-0442-17-7-1487-e5
where σ2e(τ) and σ2 are prediction error variance and climatological variance, respectively. In Figs. 5a,b, ϵ2(τ) is shown as a function of lead time τ and coupling parameter β. The thick 0.5 contour gives the lead times at which the signal variance and error variance are equal and the system loses its predictability. For the normal system (Fig. 5b), ϵ2(τ) only depends on the real part of the eigenvalues that determines the damping rate of the system. Because the damping rate increases as the coupling strength decreases, one expects that the predictability of the system decreases rapidly as coupling strength decreases. Indeed, Fig. 5a shows that the predictability of the system decreases nearly exponentially as the coupling strength decreases. Useful forecast skills of the normal system are limited in the strong coupled regime. For coupling strength less than 1/200 (days−1), the predictability of the system is less than 30 days.

For the nonnormal system (Fig. 5a), the normalized error variance grows at a much slower rate than the normal system, indicating an enhanced predictability due to the interference among the eigenmodes. The enhanced predictability is particularly evident for relatively short lead times, confirming the theoretical result in section 3c of Part I. Furthermore, for the system at hand, the normalized error variance growth is improved over the normal system not only for short lead times, but also for long lead times. This is shown more clearly by the difference between the normalized error variances of the two systems (Fig. 5c). From Fig. 5c it is evident that the enhanced predictability by nonnormal growth is largely confined to the weak-to-moderate coupled regimes. In these coupled regimes, the time limit of the predictability increases from a few months in the normal case to more than a year in the nonnormal case. This result suggests that the dynamics that control the system's predictability may be different in different coupled regimes. In the strong coupled regime, the system predictability depends primarily on the characteristics of the leading eigenmode, because the leading mode tends to dominate the system's time evolution. In weak-to-moderate coupled regimes, however, the interference among the eigenmodes plays a more important role and the nonnormal growth is an important contributing factor to the system's predictability.

The persistence ρ(τ) and correlation skills γ(τ) calculated according to (31) and (32) in section 3b of Part I are shown in Figs. 6 and 7. Again, these skills are computed separately for the nonnormal and the normal systems to examine the contribution due to nonnormal growth. The persistence, as expected, exhibits a damped oscillation in the strongly coupled regime. Because the oscillation characteristics are determined by the eigenvalue of the dominant eigenmode, one expects little change in the oscillatory behavior of the persistence between the two systems. Indeed, the zero contours in Figs. 6a,b are nearly identical, indicating that the two systems possess similar oscillation characteristics. However, the amplitude of the persistence between the two systems shows significant differences, the nonnormal system has larger amplitude than the normal counterpart for short lead times. This means that the persistence forecast of the nonnormal coupled system is more skillful than that of the normal system. Similar results also hold for correlation skill, as shown in Fig. 7. Therefore, we have demonstrated that for the simple TAV model the predictability does not simply come from oscillatory behavior of the dominant coupled modes. The nonnormal growth caused by the interference among the coupled modes can contribute significantly to the system's predictability.

To bring this discussion to a physically more meaningful context, we compare the predictive skills of both systems to those of the degenerate system where V(y) and β are both set to zero. As mentioned previously, the degenerate system is normal and supports no oscillatory modes. The dotted lines in Figs. 5d,e,f; 6d,e,f; and 7d,e,f show the skills of the degenerate system along with the skills of the coupled system and its normal counterpart for three selected values of the coupling parameter representing three different coupled regimes—weak [β ≥ 1/500 (days−1)], moderate [β ≥ 1/200 (days−1)], and strong [β ≥ 1/120 (days−1)]. When coupling is strong [β ≥ 1/120 (days−1)], the skills of the normal system are comparable to the skills of the nonnormal system and both are significantly above the skills of the degenerate system. For weak-to-moderate coupling strength [β ≤ 1/200 (days−1)], the skills of the normal system are comparable to or even worse than the skills of the degenerate system, while the skills of the nonnormal system remain superior to those of the two subsystems.

The above results can be understood in terms of the relationship between temporal and spatial characteristics of the eigenmodes and predictability. For a normal system, the normalized prediction error ϵ2(τ) is solely determined by the real part of eigenvalues of the dynamic operator 𝗔. Therefore, coupled dynamics can affect the predictability only by altering damping rate of the eigenmodes, not by changing the cyclic behaviors of the modes. This effect becomes dominant when the positive air–sea feedback is sufficiently strong so that the coupled system is close to being unstable. In such a strongly coupled regime, one expects the skill of the normal system to be comparable to that of the nonnormal system, and to be significantly better than the skill of the degenerate system. This is because the air–sea coupling that is absent in the degenerate system acts to reduce thermal damping in the coupled model. However, coupled dynamics can not only affect the temporal characterisitics of the eigenmodes, but can also alter their spatial structures. In particular, the nonlocal effect of the air–sea coupling can enhance the interference among the eigenmodes, which can have a significant impact on the predictability. This assertion is supported here by the finding that the skill of the coupled model is superior to the two normal systems when the feedback strength is moderate, suggesting that in this regime coupled dynamics contribute to the system's predictability primarily through enhancement of the nonnormal growth.

b. Effect of noise forcing

The predictability of the coupled system depends not only on the dynamics of 𝗔, but also on the structure of the noise forcing. Once the system's dynamic operator 𝗔 is given, the predictability is determined by the structure of the noise forcing. In section 3d of Part I, we show that there exists an optimal noise forcing fϵ2 under which the system's predictability can be maximized. Here fϵ2 is given by a generalized eigenvalue problem:
fϵ2ντfϵ2
where 𝗕(τ) = τ0 e𝗔*se𝗔s ds, ν = ϵ−2. The leading eigenvector fϵ2 corresponding to the largest eigenvalue ν defines the optimal forcing pattern that gives the smallest normalized error ϵ2.

We computed the optimal noise forcing fϵ2 for the simple coupled model. Figure 8 shows fϵ2 as a function of lead times for a moderate coupling strength, β = 1/200 (days−1). It is interesting to note how the optimal noise forcing departs, as lead time τ increases, from the stochastic optimal, which corresponds to fϵ2 at τ → 0 (Fig. 8): the maximum forcing amplitude shifts from the northern Tropics to south of the equator. This feature is further illustrated in Fig. 8b where the stochastic optimal (dashed) is contrasted to the optimal noise forcing fϵ2=0.5 at the lead time τ, where ϵ2(τ) = χ2(τ) = 0.5 (solid). It is also interesting to note how different these optimal forcing patterns are from the leading eigenmode. The former has a narrow structure with a maximum amplitude south of the equator for large lead times, while the latter has a broader dipolelike structure with a large amplitude in the northern Tropics. This difference in the structures of the eigenmode and optimal forcing reflects the nonnormality of the system, because these structures are expected to be identical only when the system is normal.

From the discussion in section 3d of Part I, we know that each of these optimal forcings gives the system a minimum error variance growth at the corresponding lead time. In other words, if one of these forcing patterns is used to force the linear coupled model, the error variance ϵ2 will be minimum at the corresponding lead time and the system will achieve its maximum predictability under any possible noise forcing. Figure 8c illustrates the normalized error variance growth as a function of lead times τ under different optimal noise forcing. As expected, the stochastic optimal (dashed) gives the best predictive skill at short lead times, but not at longer lead times. On the other hand, if the system is forced by fϵ2=0.5, the normalized prediction error (thin solid) is improved over that of the stochastic optimal for τ ≤ 200 days, but not for short lead times. The upper limit of the system's predictability under any noise forcing is obtained by minimizing the error growth at each lead time by using the optimal noise forcing at that lead time. The resultant limit is shown by the thick solid line in Fig. 8c. As can be seen, the increase in forecast skill can be substantial when a proper optimal noise forcing is used. For example, at ϵ2 = 0.5 the skill of the model increases from lead time τ = 300 (days) when the stochastical optimal is used to over 400 (days) when fϵ2=0.5 is used. Note that even this maximum prediction lead time is considerably shorter than the period of the least damped coupled mode.

c. Predictable component analysis

In practice, it is often difficult to explicitly separate the stochastic forcing from other processes in the system. In the case that both the dynamical operator 𝗔 and noise forcing 𝗙 are already given, we are interested in finding the pattern—the most predictable pattern—in the system's response that give the maximum predictability of any other patterns. The procedure of identifying such a pattern is called the predictable component analysis (Schneider and Griffies 1999). In section 3e of Part I, we introduced the concept of predictable component analysis in the general framework of linear stochastic models and discussed the relation between the predictable patterns and the empirical orthogonal functions (EOFs). Here, we apply this analysis to the response of the simple TAV model to a unitary noise forcing with a moderately strong coupling parameter β = 1/200 (days−1). As shown in section 3e of Part I, the pattern p that maximizes ν = 1/ϵ2q, is given by the leading eigenvector of the generalized eigenvalue problem:
τ−1pνp
where 𝗖(∞) and 𝗖(τ) are the climatological covariance and error covariance matrices, respectively.

Figure 9 shows p as a function of lead times. From the discussion in section 3e of Part I, we know that the most predictable pattern degenerates into the leading EOF at very short lead times. Figure 9b contrasts the leading EOF of the simple coupled system to the predictable pattern that minimizes the normalized error variance at the lead time τ of ϵ2(τ) = χ2(τ) = 0.5 (solid). Both patterns have similar structures and have an appearance of a dipole, which bears a resemblance to the least damped eigenmode in the system (Fig. 2). In fact, all the predictable patterns for lead time τ up to 700 days display a similar dipole structure to the leading EOF, indicating a weak dependence on lead times. This result indicates that the dipolelike mode is the most predictable pattern in the system.

Figure 9c shows the normalized error variances associated with the predictable patterns as a function of lead times. The upper limit of the system's predictability under the given noise forcing can be obtained by applying the predictable component analysis at each lead time (thick solid line). As expected, the normalized error growth is reduced along the direction of the predictable patterns. However, the improved error variance growth is generally modest. This result suggests that although the predictable pattern can differ from the leading EOF in theory, this difference in practice may be small. Therefore, the leading EOF not only represents the most predictable pattern within an observation dataset for short lead times, but may also give a good approximation for the predictable pattern of the governing system for longer lead times.

5. Numerical experiments

The simplicity of the TAV model permits the theoretical analyses to be carried out using stochastic calculus without discretizing in time. These solutions are exact within the accuracy of the spatial discretization and eigenvalue analysis. Such an analytical approach, however, becomes impractical for more complex systems, such as, coupled general circulation models. For these systems, the predictability study is often done by performing numerical integrations. One of the interesting questions is under what conditions the numerical solution will converge to the theoretical prediction. This issue can be conveniently addressed in the framework of the simple model, because its theoretical predictability is known.

To investigate this, we performed a numerical integration of (2) on the same spatial grid used for the theoretical analysis, using a leapfrog scheme in time with a time step of Δt = 12 h. A Monte Carlo approach was used to obtain an estimate of predictive error distribution due to sampling errors. A brief outline of the approach is as follows. First, a verification period for the prediction experiment and a sampling interval are chosen. Then, an ensemble of 1000 prediction runs was carried out for the chosen verification period, each of which is under a different noise realization. Predictive skills of each ensemble member were computed by verifying prediction against the model's own simulation using the chosen sampling interval. Finally, the ensemble mean and standard deviation of the prediction skills from the ensemble of 1000 prediction runs were computed and compared with the theoretical results.

As an example, we chose the noise forcing structure to be uniform and the coupling parameter β of 1/200 (days−1). Sensitivity studies indicate that the sampling interval does not have a profound impact on the estimate of predictability, provided that it is short enough (less than a season) to adequately resolve the temporal variability of the system. However, the length of integration or the verification period does have a significant impact on predictability estimates. Figure 10 shows numerical estimates of the system's persistence and correlation skills, as well as normalized error variance from four sets of ensemble prediction runs with verification periods of 10, 20, 50, and 100 yr and a sampling interval of 1 month. For short verification periods (10–20 yr), numerical estimates tend to deviate significantly from the means, indicating that a large uncertainty results from a small sampling size. From Fig. 10 one observes a tendency that in this case the numerical estimates generally underestimate the true predictability of the system. As the verification period increases, the spread of the numerical solutions decreases and the ensemble means approach to the theoretical results. This result suggests that the skill estimates based on short verification period must be dealt with extra caution. For example, several recent studies (e.g., Balmaseda et al. 1995) have noted that the persistence of ENSO and models' ENSO forecast skills exhibit some interesting decadal fluctuations. Given the shortness of the observed record, it is possible that these decadal fluctuations originate from the insufficient sample size associated with the short verification period.

6. Summary and discussion

We have made an attempt to illustrate the complexity of the predictable dynamics of weakly coupled systems, using a simplified coupled model of tropical Atlantic variability (TAV) as an example. Despite the simplicity of the model, we believe that the dynamical processes in it are realistic enough to shed light on some important issues concerning the predictability of coupled climate systems in general. One important result emerging from this study is that air–sea feedbacks can affect the predictability not necessarily by generating oscillatory coupled modes, but by enhancing interference among many damped coupled modes. We explored the detailed mechanism through which this active coupling affects the predictability.

For the simple TAV model, we showed that the enhanced modal interference primarily comes from the nonlocal effects associated with the active air–sea coupling. Unlike in the extratropics, the wind-induced surface flux plays an active role in the TAV system. This is supported by several recent studies that show the surface wind variability in the tropical Atlantic is strongly affected by the meridional SST gradient variability near the equator. The change in the winds can have an influence on the surface heat flux through evaporation, which in turn can further modify the underlying SST, giving rise to a positive feedback. Because the change in the wind speed depends on the SST gradient, not directly on the local SST anomaly itself, the resultant feedback tends to be nonlocal in the sense that the flux change in one location is a response of the atmosphere to SST change in other locations. It is these nonlocal effects that are highlighted in the simple TAV model and shown to play an important role in enhancing predictability. The predictability analysis of the simple TAV model indicates that the enhanced modal interference due to the nonlocal effects of the air–sea coupling can increase the predictability by nearly 50% in a weak-to-moderate coupling regime, making the skill of the coupled model prediction far greater than the persistence forecast. This means that a relatively weak active coupling can lead to a substantial increase in the predictability of a coupled system, even though the system does not support a self-sustained oscillation.

Another important factor that affects the predictability of a weakly coupled system is the structure of the noise forcing. If the system's dynamical operator 𝗔 is known, an optimal noise forcing for maximizing the predictability can be found. We have illustrated how to use this analysis to establish the upper limit of the predictability using the simple TAV model. Our results show that when the system is in a weak-to-moderate coupled regime, the predictability can be enhanced substantially by the optimal noise forcing. The upper limit of the useful predictability for the coupling parameter β = 1/200 (days−1) (determined by the lead time at which the error variance ϵ2 increases above 0.5) is about 440 days. However, even this upper limit of predictability is only a fraction of the oscillation period of the leading coupled mode. This implies that just because there is a weak oscillatory signal in the coupled system, it does not mean that the predictability of the system can be extended to the period of the oscillation. In fact, the predictability may not have anything to do with the oscillation at all. In the case of TAV, there have been suggestions of the existence of a weak decadal oscillation in SST at an 11–12-yr time scale. Assuming that this decadal variability is a manifestation of a damped coupled mode excited by stochastic forcing, as in our simple coupled model, our analysis suggests that the predictability of this phenomenon is likely to be limited to a period of a few seasons to a year.

The concept of the optimal noise forcing may also offer a different perpective on decadal climate variability. For example, let us assume, for the sake of argument, that a climate phenomenon, such as ENSO or TAV, may be described as a stochastically forced stable linear system governed by (1). In this system, the noise forcing 𝗙η can be expanded in terms of optimal noise forcings. Let us further assume that the noise statistics change from one decade to another, so that the projections of the noise onto the optimal noise forcings differ from one decade to another. Our results would then suggest that the predictability of the system would be different for these decades, simply due to changes in the characteristics of the noise forcing. In particular, maximum predictability (or minimum normalized error growth) will be attained in a decade when the noise forcing projects more favorably onto the leading optimal noise forcing fϵ2. Note that in this scenario the decadal change in the system's predictability is not associated with any change in the dynamic operator 𝗔, but only with the change in the spatial structure of the noise forcing.

Using the simple TAV model, we have also illustrated how to identify the most predictable pattern in the system. We found that when the system is forced by a unitary noise forcing, the most predictable pattern has a similar structure to the leading EOF, both resembling the leading eigenmode, which has a dipolelike structure. Therefore, in this simple system, the dipolelike SST variability is not only the most dominant mode, but also the most predictable.

Insofar as TAV is concerned, a number of important factors that have been ignored in the simple model analysis can potentially affect the predictability of TAV. First, as already mentioned in the introduction, the variability in the tropical Atlantic involves both an ENSO-like equatorial dynamic feedback (Zebiak 1993) and a thermodynamic feedback between interhemispheric SST and latent heat fluxes (Chang et al. 1997). The former has not been taken into the consideration in this study. The inclusion of this feedback will further complicate the dynamics of the coupled system, requiring a more complete representation of oceanic dynamics to allow for subsurface ocean adjustment. How this dynamic feedback will affect the predictable dynamics of TAV is not clear and deserves further investigation. Second, in reality the ENSO-like equatorial variability and the interhemispheric SST variability have a strong seasonal dependence. The former occurs mainly in the boreal summer and fall, while the latter occurs primarily in the boreal winter and spring (e.g., Chang et al. 2000). This seasonal manifestation suggests that different kinds of coupled dynamics may be operating during different seasons and the excitation of these modes may vary from season to season. This raises the issue of whether it is appropriate to treat the linear coupled system of TAV as an autonomous system. Future studies should extend the predictability analysis to a nonautonomous system. Finally, it has been shown that ENSO is an important influencing factor on TAV and its predictability (Enfield and Mayer 1997; Penland and Matrasova 1998). Yet, the detailed dynamics of the remote influence of ENSO on TAV's predictable dynamics have not been understood. Further studies are needed to address these issues to gain a full understanding of the predictability of TAV.

Acknowledgments

This work benefitted from useful discussions with Timothy DelSole, Cecile Penland, Michael Tippett, and Steve Zebiak. We are grateful to two anonymous reviewers for their constructive comments that helped improve the manuscript considerably. This work is supported by NOAA and NSF through research Grants NA16GP1572 and ATM-99007625. PC also acknowledges the support from the National Natural Science Foundation of China (NSFC) through Grant 40128003.

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Fig. 1.
Fig. 1.

Real (dash line) and imaginary (solid line) parts of the largest eigenvalue (sorted by real part) of the linear operator 𝗔 as a function of coupling parameter β. The system becomes unstable at approximately β = 1/112.25 (day−1), at which point the real part of the eigenvalue changes from negative values to positive values. For coupling parameter β between 1/900 and 1/112 (day−1), the dominant eigenmode shows a damped oscillation with periods greater than 6 yr

Citation: Journal of Climate 17, 7; 10.1175/1520-0442(2004)017<1487:POLCSP>2.0.CO;2

Fig. 2.
Fig. 2.

Time–latitude plots of the dominant eigenmode whose eigenvalue is shown in Fig. 1 for coupling parameter β of (a) 1/120, (b) 1/200, and (c) 1/500 (day−1). For illustration purposes, the exponentially decaying part of the solution is omitted. The contour interval is 0.05. The oscillation period is given by the imaginary part of the eigenvalue in Fig. 1. Note that the modal structure changes from a “dipole” to a “monopole” as the coupling parameter decreases

Citation: Journal of Climate 17, 7; 10.1175/1520-0442(2004)017<1487:POLCSP>2.0.CO;2

Fig. 3.
Fig. 3.

Power spectrum of (a) the coupled system and (b) the corresponding normal system as a function of coupling parameter β (see text for their definitions). The dash–dot lines indicates the period of the dominant eigenmode shown as the imaginary part in Fig. 1. The dashed line denotes the maximum spectral power density or spectral “peaks” at a given coupling parameter. Note that energy level in the full system is substantially higher than that of the normal portion of the system. A nonuniform contour interval (approx logarithmic) is used. Shading indicates that the power densities are greater than a nondimensional value of 100. The spectra for selected values of the coupling parameter, (c) β = 1/500, (d) 1/200, and (e) 1/120 day−1. In each the solid line represents the spectrum of the nonnormal system, the dashed line represents the spectrum of the normal system, and the dash–dot line represents the spectrum of the degenerate system where both V(y) and β are zero

Citation: Journal of Climate 17, 7; 10.1175/1520-0442(2004)017<1487:POLCSP>2.0.CO;2

Fig. 4.
Fig. 4.

Pseudospectra of the coupled system for coupling parameter β = 1/200 day−1. The points indicate the first 30 eigenvalues of the system. The contours indicate pseudospectra for different values of ϵ ranging from 10−12 to 10−7

Citation: Journal of Climate 17, 7; 10.1175/1520-0442(2004)017<1487:POLCSP>2.0.CO;2

Fig. 5.
Fig. 5.

Normalized error variance growth of the full system, as expected from the theory under a unitary forcing as a function of coupling parameter for (a) the full system and (b) its normal counterpart. (c) The difference between the two error variances. The error variance of each system is normalized by its own total variance. A contour interval of 0.1 is used in (a) and (b). The contour interval in (c) is 0.05. The thick contour line of value 0.5 gives the lead times when the system's signal-to-noise ratio reaches unity. Error variance of the (d) nonnormal and (e) normal and (f) the difference between the two systems for selected values of coupling parameter, β = 1/120 (solid), 1/200 (dash), and 1/500 (dash–dot) day−1. In each the dotted line shows the error variance of the degenerate system where both V(y) and β are zero

Citation: Journal of Climate 17, 7; 10.1175/1520-0442(2004)017<1487:POLCSP>2.0.CO;2

Fig. 6.
Fig. 6.

Theoretical persistence skill of the coupled system under a unitary forcing as a function of coupling parameter β for (a) the full system and (b) its normal portion. (c) The difference between the two correlation skills. A contour interval 0.1 is used in (a) and (b). The contour interval in (c) is 0.05. Persistence of the (d) nonnormal and (e) normal and (f) the difference between the two systems for selected values of coupling parameter, β = 1/120 (solid), 1/200 (dash), and 1/500 (dash–dot) day−1. In each the dotted line shows the persistence of the degenerate system where both V(y) and β are zero

Citation: Journal of Climate 17, 7; 10.1175/1520-0442(2004)017<1487:POLCSP>2.0.CO;2

Fig. 7.
Fig. 7.

Theoretical correlation skill of the coupled system under a unitary forcing as a function of coupling parameter β for (a) the full system and (b) its normal portion. (c) The difference between the two correlation skills. A contour interval 0.1 is used in (a) and (b). The contour interval in (c) is 0.05. Persistence of the (d) nonnormal and (e) normal and (f) the difference between the two systems for selected values of coupling parameter, β = 1/120 (solid), 1/200 (dash), and 1/500 (dash–dot) day−1. In each the dotted line shows the persistence of the degenerate system where both V(y) and β are zero

Citation: Journal of Climate 17, 7; 10.1175/1520-0442(2004)017<1487:POLCSP>2.0.CO;2

Fig. 8.
Fig. 8.

(a) Optimal noise forcing structure as a function of lead time τ. The pattern has been normalized to unit variance at each lead time. The horizontal line marks the lead time at which the signal-to-noise ratio is equal to unity, i.e., ϵ2(τ) = χ2(τ) = 0.5. (b) Stochastic optimal (dashed) and the optimal noise forcing fϵ2=0.5 (solid). (c) Error variance ϵ2(τ) growth under stochastic optimal (dashed), the optimal noise forcing fϵ2=0.5 (thin solid), and the optimal noise forcing which maximizes the predictictability at each lead time (thick solid)

Citation: Journal of Climate 17, 7; 10.1175/1520-0442(2004)017<1487:POLCSP>2.0.CO;2

Fig. 9.
Fig. 9.

(a) The first predictable pattern as a function of lead time τ. The pattern has been normalized to unit variance at each lead time. The horizontal line marks the lead time at which the signal-to-noise ratio is equal to unity, i.e., ϵ2(τ) = χ2(τ) = 0.5. (b) The EOF that corresponds to the first predictable pattern at lead time τ = 0 (dashed) and the first predictable pattern at lead time τ of unity for the signal-to-noise ratio (solid). (c) Error variance ϵ2(τ) growth associated with the EOF (dashed), the first predictable pattern of unity signal-to-noise ratio (thin solid), and the patterns that minimize the error growth at each lead time (thick solid)

Citation: Journal of Climate 17, 7; 10.1175/1520-0442(2004)017<1487:POLCSP>2.0.CO;2

Fig. 10.
Fig. 10.

Numerical estimates of the model's predictive skills using a Monte Carlo simulation with a verification period of (a) 10, (b) 20, (c) 50, and (d) 100 yr. Each set of experiments consists of 1000 simulations. The dashed lines represent the ensemble averages of predictive skills and the shading indicates standard deviations within the ensemble. The solid lines represent the theoretically expected predictive skills. It is evident that the numerical estimates converge to the theoretical solutions as verification period increases. The numerical experiments were conducted using a uniform noise forcing with coupling parameter β = 1/200 (day−1)

Citation: Journal of Climate 17, 7; 10.1175/1520-0442(2004)017<1487:POLCSP>2.0.CO;2

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