Initialvalue predictability of prominent modes of North Pacific subsurface temperature in a CGCM
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Abstract
Three 40member ensemble experiments and a 700 year control run are used to study initial value predictability in the North Pacific in Community Climate System Model version 3 (CCSM3). Our focus is on the leading two empirical orthogonal functions (EOFs) of subsurface temperature variability, which together produce an eastward propagating mode. Predictability is measured by relative entropy, which compares both the mean and spread of predictions of ensembles to the model’s climatological distribution of states. Despite the fact that EOF1, which is structurally similar to the observational Pacific Decadal Oscillation (PDO), has pronounced spectral peaks on decadal time scales, its predictability is less than 6 years. Additional predictability resides in the tendency of EOF1 to evolve to EOF2, primarily through simple advective processes. The propagating mode represented by the combination of EOF1 and EOF2 is predictable for about a decade. Information in both the mean and spread of predicted ensembles contribute to this predictability. Among the leading 15 EOFs, EOF1 is the least predictable mode in terms of the rate at which the corresponding principal component disperses in the ensemble experiments. However, it can produce enhanced predictability of the whole system by inducing EOF2, which is one of the two EOFs with the slowest dispersion rate. The first two EOFs can also enhance the ensemble mean (or “signal”) component of predictability of the entire system. For typical amplitude initial states, this component contributes to predictability for about 6 years. For initial states with unusually high amplitude projections onto these two EOFs, this contribution can last much longer. The major findings from the three ensemble experiments are replicated and generalized when the initial condition predictability for each of many hundreds of different initial states is estimated. These estimates are derived from the behavior of a linear inverse model (LIM) that is based on the intrinsic variability present in the control run.
Keywords
Empirical Orthogonal Function Pacific Decadal Oscillation Relative Entropy Subsurface Temperature Kuroshio Extension1 Introduction
While models, observations and assimilation methods have now progressed to the stage that attempts are being made to carry out forecasts on decadal timescales (Smith et al. 2007, Keenlyside et al. 2008), it remains unclear to what degree information in the initial state can add to the skill of such forecasts. For the climate system is chaotic and its evolution could be so sensitive to inevitable errors in the initial state that information residing there is lost after only a few years. If so, there is no point in trying to predict beyond this range unless one is also taking into account changing external forcing.
In this paper, we contribute to the quantification of the initialvalue predictability of the coupled atmosphere–ocean system by concentrating on the predictability of prominent intrinsic modes of decadal variability. We focus on these large scale features because they have high amplitude and thus have the potential to impact weather and climate events that affect society. Furthermore, in studies of weather prediction (e.g. Tribbia 1988 and Branstator et al. 1993) and intraseasonal variability (e.g. Newman et al. 2003a), prominent modes have been found to be more predictable than other features, and we are interested in whether this may also be true for decadal timescales.
Given the short climate record, the only way to estimate decadal predictability limits resulting from initial state uncertainty is to use models. Most previous modelbased studies have focused on the North Atlantic, particularly the Atlantic meridional overturning circulation (AMOC; Griffies and Bryan 1997a, 1997b; Collins 2002; Collins and Sinha 2003; Pohlmann et al. 2004; Collins et al. 2006b). Many studies concur that the AMOC is potentially predictable a decade in advance, but the characteristics of the AMOC, including its predictability limit, vary from model to model (Hurrell et al. 2009). And it remains controversial whether the AMOC can bring decadal predictability to the surface climate.
In the North Pacific, the thermohaline circulation is too weak to be responsible for the pronounced decadal variability observed in sea surface temperature (SST). Instead, about onethird of the SST decadal variability is associated with the “Pacific Decadal Oscillation”, or PDO (Mantua et al. 1997), a mode captured by the first empirical orthogonal function (EOF) of SST. Because of its importance, much of our investigation is concentrated on the predictability of the PDO. Several mechanisms have been proposed for the PDO (Miller and Schneider 2000; Seager et al. 2001), and they have different implications for its predictability. Latif and Barnett (1994, 1996) hypothesized that enhanced predictability can result from the oscillatory nature of the PDO, which according to their analysis relies on coupling between the subtropical ocean gyre circulation and the Aleutian low pressure system. On the other hand, driving of the ocean by white weather noise alone, without twoway coupling, can produce decadal variability in the ocean (Frankignoul and Hasselmann 1977; Frankignoul et al. 1997). A third scenario that has been suggested emphasizes the ocean circulation and slow dynamical oceanic modes that contribute to a reddening of the SST signals, even when the oceantoatmosphere feedback is weak (Saravanan and McWilliams 1998).
Predictability from initial conditions (what Lorenz (1963) refers to as “predictability of the first kind”) is measured by determining for how long the predicted distribution of an ensemble of similar initial states is distinguishable from the climatological distribution. Hence asking how predictable the PDO is on decadal time scales is different from asking how much variability it has on these scales. Even if its temporal spectrum has a decadal peak, if fast growing errors resulting from initial uncertainties quickly conceal the PDO signals in forecasts, then the PDO is not predictable. Note measuring predictability is different from measuring a quantity that some refer to as “diagnostic potential predictability” (Boer 2000). This quantity measures the variability on, say, decadal time scales that is in excess of that expected if the part of the spectrum for frequencies lower than annual were white. The straightforward way to quantitatively measure predictability is to use ensemble experiments with perturbed initial conditions under the “perfect model” assumption. Most studies use the ensemble spread as measured by standard deviations to quantify its limit, and predictability is lost when the spread is indistinguishable from the climatological spread.
Earlier studies have found that the answer to whether there is decadal predictability in the North Pacific varies with the analysis method and the variable that is considered. If one assumes there is a correspondence between diagnostic potential predictability and predictability, some modeling studies find there is decadal predictability of SST in some regions of the North Pacific (Boer 2004). On the other hand, others using stochastic modeling methods to estimate predictability, have found that SST is predictable for only 1–2 years in both the observational record (Alexander et al. 2008) and AOGCMs (Newman 2007). SST anomalies can persist into the following winter due to the “reemergence mechanism”, where the anomalies stay below the mixed layer in summer and reentrained into the mixed layer in the following fall and winter (Deser et al. 2003). But even with the influence of this effect, most studies conclude North Pacific SSTs are unpredictable on decadal time scales.
This result indicates there may exist decadal predictability in the subsurface temperature in the North Pacific in CCSM3, but it does not explain what processes and structures this predictability is associated with. In particular it does not address whether it may arise from some dynamic modes such as the PDO. With this figure in mind, in this paper we will address the following questions: to what extent is there predictability in the prominent modes in the subsurface temperature? How much do the prominent modes contribute to the predictability of the entire North Pacific region?
To answer these questions we will first identify strong modes of North Pacific subsurface decadal variability in CCSM3 using EOF analysis. It turns out that the leading EOF is similar to the structure of the PDO. Then we will quantify the predictability of these patterns using both ensemble experiments, including the one used to produce Fig. 1, as well as a 700 year control run of that model. In addition to focusing on the role of intrinsic modes in decadal prediction, this work expands on previous studies by examining the upper 300 m temperature instead of SST, and by analyzing a model whose predictability has not been previously quantified. Moreover, instead of treating the PDO as a standing mode represented by a single pattern, we find the leading two EOFs in upper 300 m temperature in CCSM3 represent different phases of a propagating mode making it more physical and instructive to consider the predictability of this pair of structures in combination than individually. One method we employ that facilitates the quantification of the predictability limit of a mode of this type is to use relative entropy (Kleeman 2002) from information theory.
To describe our investigation, this paper is organized as follows. Section 2 introduces CCSM3, the ensemble experiments, and analysis methods. The prominent modes in the subsurface temperature in CCSM3 as well as the processes that contribute to their behavior are described in Sect. 3. Initial value predictability of the dominant modes is estimated using three ensemble experiments in Sect. 4. In order to test whether the ensemble experiment results reflect general properties of CCSM3, we see whether similar predictability properties for these modes are implied by the intrinsic variability of that model in a long control run. These predictability properties are determined by construction and analysis of a linear inverse model in Sect. 5. How the leading modes contribute to the predictability of North Pacific subsurface temperatures in general is discussed in Sect. 6, followed by a summary and discussion of our study’s implications in Sect. 7.
2 Model, experiments and analysis methods
2.1 CCSM3 and experiments
CCSM3 is a fully coupled model that includes four components: atmosphere, ocean, land, and sea ice (Collins et al. 2006a). These components are linked via a flux coupler and no flux corrections are employed. Alexander et al. (2006) have compared the extratropical atmosphere–ocean variability over the Northern Hemisphere in a CCSM3 control run to observations. In the North Pacific, the model simulates the PDO reasonably well. The largest discrepancy is that the PDO lacks the connection to tropical Pacific SST that is seen in the observations. Since tropical Pacific SST can strongly influence the atmospheric circulation of the midlatitudes through the atmospheric bridge (Alexander et al. 2002), the model may underestimate climate anomalies associated with the PDO over the surrounding continents. Because the main purpose here is to quantify the predictability of the prominent modes in the North Pacific rather than their impacts on variables over land, such model weaknesses may not be important. On the other hand, since apparent tropicalmidlatitude interactions implied by the observational record on decadal time scales are not well understood it is possible that these interactions have an influence on the predictability of the PDO (Newman et al. 2003b) that is not present in the simulations we study here.
To assess the predictability of prominent modes in the North Pacific, we assume CCSM3 is a perfect surrogate for nature and perform three 40member ensemble experiments using a configuration with a T42 atmosphere and a nominal 1° ocean. Each of the three experiments has a different initial state while the realizations in each experiment differ from each other in one of two ways. Ensemble I is branched from January 1st of year 2000 of a 20th century historical run. Its 40 initial conditions are identical for land, ocean, and sea ice gridpoints, but atmospheric initial conditions are unique to each realization; they are taken from different days in December 1999 and January 2000 of the historical run. Ensembles II and III are branched from two of the realizations of Ensemble I in year 2008. In the latter two ensembles, all 40 members share the same initial states for all four model components; each realization is distinct only because the solar constant is slightly different in each one. The realizationtorealization variations in the solar constant are so small as to produce no appreciable effects on the climate of each realization. On the other hand, after one time step these variations have produced very small variations in the atmosphere and thus act to set each realization off on a different trajectory. Hence the solar constant perturbations serve the same purpose as would small perturbations to the initial atmospheric state.
In our study of the predictability of prominent North Pacific modes, an inevitable question is whether the predictability limit of these modes is sensitive to their initial amplitudes. This question guides our choice of initial ocean states. As mentioned in the introduction and as described in detail later, the mode our study primarily focuses on is defined by the leading two EOFs of subsurface temperature. The initial state for Ensemble I has weak projections on both these patterns. By contrast, the initial state for Ensemble II is chosen because it features a very strong projection onto EOF1 but has a very weak projection on EOF2, and the initial state of Ensemble III has a very strong projection on EOF2 but a very weak projection on EOF1. Ensemble I is integrated for 62 years, but in results not presented here, we have found that if there is any predictable signal beyond year 20 it is very weak. Therefore, we only consider the first 20 years of that ensemble, and Ensembles II and III are integrated for only 20 years.
Although our study focuses on initialvalue predictability of intrinsic modes, A1B scenario forcing (Meehl et al. 2006) has been included in all three ensembles so that these experiments can also be used for assessing predictability resulting from this external forcing (Lorenz’s (1963) “predictability of the second kind”). In a followingup study, we will compare predictability of both kinds using these ensemble experiments. In the present study, we assume intrinsic variability is independent of the trend caused by the external A1B forcing trend. This allows us to concentrate on intrinsic variability by first calculating the 40member averaged linear trend in Ensemble I during 1999–2061 at each grid point, and then removing this trend from each realization in all three ensembles. The residuals are the variability whose predictability we examine.
In addition to the three ensembles, we employ a 1000 year control run (Bryan et al. 2006) for learning about the model’s intrinsic modes, for constructing the inverse model describe below, and for assessing statistical significance. It is identical to the model used for the ensemble experiments except the forcing is set to the conditions for 1990. Our study only uses the last 700 years of the control run to avoid the years when spinup occurs.
Our analysis uses values of SST and meanupper300 m temperature in the North Pacific region defined by (20°N–65°N, 120°E–110°W). We average the ocean temperature in the top 16 layers of the ocean model and then regrid it from the native ocean grids to the atmospheric model’s T42 grid. All results we present are derived from annual mean data. We have also tested the use of DJF means, and it does not affect the main results of our study.
2.2 Relative entropy
In information theory, R is called the relative entropy (Kleeman 2002). Relative entropy can be thought of as being a distance between distributions e and c though strictly speaking it is not a distance in the mathematical sense. For predictions of a Markov process, R always decreases monotonically with time provided complete state vectors are employed. When R asymptotes to a value of zero, it indicates that the two distributions are identical, and the initialvalue predictability is lost.
Later, we will refer to the first two terms minus n as the dispersion component and the third term as the signal component of the relative entropy.
In our study, we have 700 samples from the control run to estimate the climatological distribution, but only 40 samples for predictions. The limited sample size of the prediction ensembles induces an error in the estimation of R. Because R is positive definite this error has a bias, and R asymptotes to a nonzero value when the predictability is lost. We estimate this value by calculating R from many samples of 40 randomly chosen states from the control run.
Relative entropy has several advantages compared to a conventional indication of predictability based only on the spread of a single variable (e.g. the ratio of predicted to climatological variance). First, it can take into account how both the predicted mean and spread compare to the corresponding climatological quantities. Clearly, this is a worthwhile attribute since there is still useful information if a prediction indicates a shift of the mean, even if it has the same spread as climatology. Second, it can quantify predictability of a system that has more than one degree of freedom. For our study, this attribute is important because we wish to consider predictability of modes that have more than one degree of freedom.
2.3 LIM
The CCSM3 ensemble approach has one important limitation, namely, predictability limits revealed by an ensemble experiment may be only valid for the particular ocean/land/ice initial states used in the experiment. In order to test whether the results we find in our ensemble experiments reflect more general predictability properties of CCSM3, we construct a linear inverse model (LIM) based on the control run. The resulting LIM should approximate the dynamical properties of CCSM3 provided the underlying assumption of the LIM is valid. This assumption is that the system being represented can be well approximated by a linear system driven by Gaussian white noise. Once we have approximated CCSM3 in this way, we can estimate its predictability limits for very many initial states.
We measure the predictability of a distribution of LIM forecasts just as we do a CCSM3 ensemble, namely in terms of relative entropy using the mean and standard deviation from Eq. 6 and 7. One property of LIM forecasts, as with any linear stochastic oscillator, is that the noise characteristics are independent of the initial state. Therefore, as can be seen in Eq. 7, the covariance structure of a forecast is not affected by the initial state. Hence only the signal component of relative entropy in Eq. 2 varies with the initial condition, and an initial state with a particular structure will have higher predictability the larger its amplitude is.
In order to reduce data processing and to exclude patterns of variability whose amplitudes are weak, we use the leading 15 EOFs in the upper300 m temperature in the North Pacific to represent the entire field when defining our LIM. Together these 15 explain about 85% of the total variance. Just as Alexander et al. (2008) and Newman (2007) found when constructing LIMs for observed Pacific SST, we find the resulting LIM is appropriate for North Pacific subsurface temperature variability in CCSM3 and approximates its behavior well. One indication of this is that when we generate our LIM with \( \tau_{0} = 1 \)year, the resulting propagator Eq. 6 explains 66% of yearly increments and 41% of 2 year increments while the corresponding values for persistence forecasts are 49% and −6%, respectively. Moreover, other conditions that a LIM should meet (Penland and Sardeshmukh 1995) if the assumptions it is based on are valid are also satisfied. The Euclidean norm of B is insensitive to variation of\( \tau_{0} \) in the 1–3 year range, and the eigenvalues of Q are all positive. Additional indications of the appropriateness of the LIM are presented in Sect. 5 where we find it can reproduce the basic predictability characteristics of the three CCSM3 ensemble experiments of our study.
3 Prominent modes
To identify prominent modes of subsurface variability that are intrinsic to CCSM3, we apply EOF analysis to the North Pacific in the control integration. Any one method of identifying modes is not necessarily adequate for all situations, but as seen below, EOFs are effective at identifying the mode of North Pacific variability that we are most interested in, the PDO. When we calculate EOFs for the average temperature in the upper 300 m, we find the first two EOFs together explain about 40% of the total variance, and they are well separated from the other EOFs (North et al. 1982). We mainly focus on the characteristics of these two EOFs. For comparison, we also examine the first two EOFs of SST in the same domain because this is the variable on which many studies have concentrated.
3.1 The first two EOFs
In contrast to EOF1, EOF2 is very different for SST and subsurface temperature (Fig. 2, right panels). The SST pattern is concentrated in high latitudes while the subsurface pattern consists of features in midlatitudes, including two lobes that resemble eastward shifts of features in EOF1. The subsurface temperature PC2 has wide frequency peaks ranging from 10 to 20 year periods (Fig. 3, bottom middle) but only the peak at 12 years is present in both dataset halves. PC2 for SST has no distinct peaks in this range.
To determine whether the EOF patterns may be two phases of a time evolving feature, we carry out a coherency analysis. While the leading two PCs of SST are related at only the 30 and 10 year periods (Fig. 3, right panels), the two PCs of subsurface temperature have significant coherency at almost all frequency bands, and the coherency is much stronger than that between the two PCs in SST. The tight relationship between the two PCs in the subsurface temperature suggests that they correspond to a time dependent physical mode. Taking into consideration their power spectra, this physical pair has a preference for variability in the 10–30 year range.
3.2 Leading propagating mode
We also apply CEOF analysis to bandpass filtered SST to detect the dominant propagating mode at the surface. Interestingly, both the real and imaginary parts of SST CEOF1 are similar to the corresponding components of CEOF1 in the subsurface temperature, suggesting that there are consistent changes in SST associated with the leading propagating mode in the subsurface temperature. In agreement with this interpretation, projections of bandpass filtered fields onto the real part of subsurface CEOF1 and onto the real part of SST CEOF1 have a correlation coefficient of 0.57 as do projections onto the imaginary parts of the CEOFs. The real part of SST CEOF1 also looks like SST EOF1, but the imaginary part is completely different from the SST EOF2, indicating that the leading propagating mode is more prominent in the subsurface temperature than at the surface. In fact, we find in results not shown, that the imaginary part of SST CEOF1 projects most strongly onto the fifth EOF of SST, a pattern that represents only about 4% of SST variability. Hereafter, we focus on the prominent modes in the subsurface temperature only, but we will examine whether their expression in SST, as suggested by the above results, is significant. Since the first two subsurface EOFs are similar in structure to CEOF1, for the sake of simplicity, we use the leading two EOFs to define the leading propagating mode in the remainder of our study.
3.3 Composite and heat budget analysis
Using the phase of the subsurface temperature, we also generate composites of SST and sea level pressure (SLP) (Fig. 6, right panels). The structure of the SST anomalies agrees with the subsurface temperature anomalies at all four phases, confirming that the leading propagating mode in the subsurface temperature is associated with consistent temperature anomalies at the surface. Interestingly, there are systematic changes in SLP as well. Positive SLP anomalies prevail in the North Pacific at phase 0°. Thereafter, the high pressure center moves slightly northwestward and negative anomalies appear in midlatitudes and propagate slowly northward. A previous study has found evidence for twoway coupling between SST and the atmospheric circulation on the decadal time scale in CCSM2 (Kwon and Deser 2007). Here we do not attempt to establish cause and effect but rather simply point out there is a linkage between the subsurface propagating mode and SLP. Although the amplitude of the SLP anomalies is relatively weak (about 0.5 hPa at the center) compared to interannual variability (~3 hPa), the persistence of the SLP anomalies may cause a significant impact on other surface variables that are of direct relevance to humanity.
Examining the contributions from the three terms we find the horizontal convergence of temperature flux (Fig. 7c) plays the dominant role. We further decompose this term into horizontal temperature advection \(  \left( {u{\frac{\partial T}{\partial x}} + v{\frac{\partial T}{\partial y}}} \right) \)and the horizontal convergence term \(  T\left( {{\frac{\partial u}{\partial x}} + {\frac{\partial v}{\partial y}}} \right) \). Again these are calculated for each model layer and then depth averaged. We find both terms are important to horizontal convergence of temperature flux (Fig. 7b) over the Kuroshio Extension region (figure not shown). In most of the North Pacific basin away from this region, horizontal advection dominates (figure not show). To determine whether advection by the mean currents is primarily responsible for this advection, we recalculate the horizontal advection term with the currents replaced by their climatological mean value (Fig. 7f). Though this component of advection is important, major features in Fig. 7c, particularly in the Kuroshio Extension region, are not reproduced by this simplification. This implies that perturbation currents cannot be ignored and raises the possibility that atmospheric anomalies are involved in producing the eastward propagation of the mode.
Unlike horizontal convergence of temperature flux, the temperature flux at 300 m depth (Fig. 7e) has the opposite sign of the total temperature change (Fig. 7d) in most regions, except at the beginning of the Kuroshio Extension and in the eastern Pacific. Therefore, it is not the main cause for the eastward propagation. The net surface heat flux (Fig. 7a) does contribute positively to the eastward propagation, especially in the Kuroshio Extension, but overall, the eastward propagation is largely driven by horizontal advection. We also calculate the composite for the residual in Eq. 8, and it explains the difference between Fig. 7d, b. It is smaller than the contributions from surface heat flux and the advection terms. Given the fact that the analysis does not rule out the possibility that atmospheric processes, with their high variability, are important contributors to the evolution of the propagating mode associated with the model PDO, prospects for it having long predictability are diminished.
4 Predictability of the leading EOF modes in the three ensembles
Having found nearly 40% of the interannual subsurface variability is composed of a pair of patterns that act together, next we examine initialvalue predictability of this pair in the three ensembles. This involves examining the PCs resulting from projecting the intrinsic component of temperature (cf. sect. 2) onto control run EOF1 and EOF2. Despite the common perception that ensemble spread is the main determinant of predictability, we find both the mean and spread make significant contributions.
As for the ensemble spread (Fig. 8c, d), in all three ensembles during the first 8 years the standard deviation is less for normalized PC2 than for normalized PC1; it takes about 5 years for the standard deviation of PC1, and 8 years for that of PC2, to exceed one standard deviation of the control run. So from the standpoint of spread, predictability of PC2 is lost much later than for PC1. Another interesting feature is that the spread in all three ensembles grows at a similar pace, despite the large differences among the initial conditions. This latter fact will be important in the next section. When considering Fig. 8, it is important to recognize that the range at which the standard deviation of an ensemble exceeds the climatological standard deviation is only a rough indication of when predictability is lost. For an infinite ensemble the approach is asymptotic and so the predicted standard deviation is never greater than the climatological standard deviation, while for a finite ensemble sampling effects can erroneously produce an apparent convergence of the two distributions before it has actually taken place.
In both Ensemble I and III, where the initial projection onto PC1 is small, PC1 is predictable for about 4 years, and the predictability mainly comes from the dispersion component (Fig. 9, left column). PC1 is more predictable in Ensemble II due to large contributions from the signal component (Fig. 9, left middle). The total relative entropy does not drop to the threshold level until year 6 when the ensemble mean drops to zero. From year 6 to year 20, relative entropy stays above the threshold as the mean PC1 remains different from climatology.
For all three ensembles, PC2 is more predictable than PC1 and the enhanced predictability comes from two sources. First, the dispersion component of relative entropy for PC2 decays more slowly than for PC1, which is consistent with Fig. 8c, d. Second, when the prediction starts with a strong PC1, the initial signal propagates to PC2 (as in Ensemble II), but a strong initial PC2 signal does not always produce a strong PC1 (as in Ensemble III). The combination of these two factors is that PC2 is predictable for 7–10 years in our experiments (Fig. 9, middle panel).
We have argued that EOF1 and EOF2 are dynamically linked through advection, so it makes sense to consider the predictability of PC1 and PC2 simultaneously. This is done in the right column of Fig. 9, which displays bivariate relative entropy for these two quantities. Again we see predictability in the 7–10 year range. Ensemble I benefits from contributions from both PCs. Ensemble II has the longest predictability with our predictability criterion even being met at year 20, but keep in mind that the strong initial projection onto EOF1 that is probably responsible is a very rare (99th percentile) event. Ensemble III is not especially predictable in spite of a strong initial anomaly, presumably because PC2 tends to simply decay. We also note that though the initial projections onto EOF1 and EOF2 are weak in Ensemble I, they stay predictable for approximately as long in this experiment as in Ensemble III. This suggests that other system components can influence the evolution of these structures.
5 Predictability estimated from a linear stochastic model
As explained in Sect. 2, we have constructed a LIM that approximates CCSM3. In that section we explained why it is reasonable to think that our LIM behavior should be a good approximation to CCSM3 subsurface temperature behavior. Further evidence of its validity is the fact that we found in Sect. 4 that there is no obvious dependence of the rate at which PCs disperse on the particular initial conditions employed in our three ensembles. Recall from Sect. 2 that this is a property that one expects to be true for systems that satisfy the assumptions that LIMs are based on. As a final test of the appropriateness of the LIM for our predictability investigation, we evaluate whether it can reproduce major predictability characteristics of the first two PCs seen in the three CCSM3 ensemble experiments.
Next, we consider cases with strong PC2 initial conditions. We select the 50 states from the control run with the strongest PC2, and repeat the above LIM simulation and analysis procedures. When we do this we confirm our main result concerning this pattern, namely that there is little if any tendency for initial information in PC2 to transfer to PC1 (Fig. 11, middle panels). There are cases when PC1 contributes to predictability but this happens early in the integrations and so is not a result of propagation.
Finally, to examine predictability from a still richer distribution of initial states, we make LIM predictions using each annual mean in the control run as an initial state. Averaged relative entropy and its components derived from predictions from the 700 initial states is shown in the bottom panel of Fig. 11. It suggests that at the start of the prediction, mean and spread contribute about equally to predictability. But after 3 years for PC1, and 6 years for PC2, predictability mainly comes from signals. Interestingly, the envelopes of relative entropy in these diagrams indicate that predictability of both PC1 and PC2 may be enhanced by the presence of initial anomalies in subsurface structures other than EOF1 and EOF2. This can be seen, for example at year 6 of PC1, where the most predictable cases out of the 700 tested have relative entropy that is 28% larger than any of the cases with strong PC1 or PC2.
6 Comparison to other EOF modes
After having quantified the predictability limits of the most prominent patterns, now we investigate whether and how much the presence of the two leading EOFs enhances predictability of the North Pacific subsurface temperature. This involves comparing their predictability attributes to those of other major patterns of variability. Again we consider that part of subsurface temperature that is represented by its first 15 EOFs, and we examine each of the CCSM3 ensembles as well as the 700 LIM solutions used in Sect. 5.
One needs to be cautious of a property of relative entropy when interpreting the top row of Fig. 13. Although relative entropy can measure predictability of a system consisting of several variables, contributions from these variables are given equal weight, as indicated by the normalization of the ensemble means in Eq. 2. And the more variables one considers, the larger the relative entropy will be, so the percentage contributions presented above would be different if we had used a different number of EOFs to describe the temperature field. A modification that is sometimes used to the definition of entropy can avoid this situation. This modification (Karmeshu and Pal 2003) corresponds to including a weighting function in the integrand of Eq. 1 that is a function of x. Following this idea, we have recalculated the top row of Fig. 13 with the contribution to the signal term from each PC weighted by the variance of that PC in the control. This has the effect of removing the division by \( \sigma_{c}^{2} \) in Eq. 2. In doing this, we are reasoning that information associated with a variable that tends to have high amplitude is more useful than information associated with low amplitude variables. The results are shown in the bottom row of that figure. As one would expect, with this weighting the contribution to the signal from PC1 and PC2 becomes much more important. Especially in Ensemble II and III, but even for the average LIM case, for at least 6 years the contribution from these two components is comparable to that from the other 13 combined. A second finding of interest that comes from using relative entropy weighted in this way is contained in the grey curve in the bottom panels of Fig. 13. It shows the signal component of weighted relative entropy if one uses 20 EOFs to represent the subsurface temperature state. In all four panels the signal is almost the same as that for 15 EOFs suggesting that when employing 15 EOFs we have included essentially all of the important contributions to information content.
7 Concluding remarks

For average amplitude events, EOF1 (the PDO) is predictable for less than 6 years.

In terms of ensemble dispersion EOF1 is the least predictable of the leading 15 EOFs while EOF2 is one of the two most predictable.

The leading two EOFs together produce an eastward propagating mode driven by horizontal advection. This mode is not oscillatory; EOF1 tends to evolve to EOF2 but it is much less likely that EOF2 will evolve to (minus) EOF1.

For initial conditions with average amplitude the propagating mode is predictable for about a decade. Higher predictability results from events beginning with large amplitude PC1 than events beginning with large amplitude PC2.

For the leading mode information in the ensemble mean tends to be longer lasting than information in the spread about the mean. On average the ensemble spread is indistinguishable from the climatological spread within 5 years while the mean signal lasts nearly a decade. For events with unusually high initial amplitude the mean signal can last even longer.

The predictability of the leading propagating mode appears to make no larger contribution to basin wide predictability than other patterns of variability from the standpoint of information. But when one considers the high average amplitude of this mode, then its impressive contribution to useful forecasts becomes apparent.
An important lesson from this study is that the intrinsic time scale of a pattern is not necessarily a good indicator of its predictability. We find that PC1 of subsurface temperature has pronounced variance on decadal time scales, but its averaged predictability limit is less than 6 years, perhaps because the atmosphere may be an important contributor to its maintenance and evolution. By contrast, PC2 has an average predictability limit of about 8 years even though the peaks in its spectrum are at somewhat higher frequencies than those for PC1. This contrast is also seen in the characteristic times of these two PCs: PC1 amplitudes are reduced by a factor of e after 4.5 years while it takes only 3.6 years for such a reduction in the more predictable PC2. The unreliability of time scale as an indicator of predictability is even more dramatic when we consider EOF5. It is the most predictable among the leading 15 EOFs with regard to dispersion rate, yet its edamping time is 3.5 years. The leading EOF of a geophysical field often possesses the longest intrinsic time scales, and it has been found to have enhanced predictability in studies of weather prediction and intraseasonal variability. Such characteristics have partly motivated us to focus on the leading prominent modes as a first step toward quantifying predictability in the North Pacific. But our results here provide a counterexample to a correspondence between time scales of a pattern and its predictability and indicate one must take a prognostic approach in order to get an accurate estimate of predictability limits.
A second clear lesson from our investigation is the need to allow for time evolving modes when examining predictability. EOF1 appears to have low predictability when considered in isolation. But this conclusion changes when one takes into account its tendency to evolve to EOF2, which is one of the most predictable patterns. Interestingly it is also apparent that the initial phase of such time dependent modes must be accounted for when quantifying their predictability.
We are uncertain which physical process determines the 20 year dominant frequency peak of the prominent mode captured by EOF1 and EOF2. Our result that the dominant eastward propagation of this mode is mainly caused by horizontal advection does not eliminate the possibility suggested by some studies (Jin 1997; Qiu 2003) that baroclinic Rossby waves resulting from ocean–atmosphere interactions may play a role in timescale selection of prominent North Pacific modes. In results not displayed in this paper, we have found suggestions of westward propagating features in sea surface height to the north of the eastward propagating features we have focused on. In addition, we find hints of anomalies propagating southwestward along the subtropical subduction pathway in composites of subsurface temperature, but they don’t propagate all the way to the western boundary. Our main point, from the perspective of predictability, is that in CCSM3 westward propagating features are not prominent enough to sustain full oscillations as exemplified by the fact that in the CCSM3 ensemble experiments and in the LIM, EOF2 does not necessarily lead to growth of EOF1.
A third important lesson from our work is that both the mean and spread of an ensemble should be taken into account when assessing initialvalue predictability. A common question for predictability studies is whether there exist some initial ocean states that are more predictable than others. In contrast to common perceptions that ensemble spread is the primary factor that distinguishes the most predictable situations (Palmer 1993), all three of our ensembles show similar rates of spread; the longest predictability is found in the ensemble with an especially strong initial anomaly that leads to a long lasting mean signal. This behavior resembles the predictability characteristics of a stochastically forced damped linear system (Kleeman 2002). The fact that we were able to replicate the statistics of the prominent propagating mode in the three ensembles rather well by a LIM further implies that the PDO in CCSM3 may to a large extent be approximated as a stochastic, damped mode.
Beyond these results and their implications, our work adds to previous predictability studies in the following ways: First, we focus on subsurface temperature. Our results suggest that the subsurface propagating mode, which has decadal predictability, is associated with signatures in SST and atmospheric surface pressure. This finding opens the possibility that filtering surface conditions to retain anomalies that are associated with predictable subsurface structures might be a way of isolating those components of surface conditions that are most predictable on decadal time scales. Second, our predictability limit is estimated using relative entropy, which can assess the forecast range at which information present in the initial condition is lost based on all aspects of a predicted distribution. An additional benefit of relative entropy is that it can measure information flow in a multivariate distribution making it ideal for considering modes with more than one spatial degree of freedom. Methodologies based only on isolated geographical locations or single patterns, e.g. univariate AR1 modeling, would be misleading under these circumstances.
Though we believe the various implications of our work are worthwhile, it is best to also keep in mind various limitations to the conclusions that should be drawn from our results because of our experimental design and model errors. Perhaps paramount among these is that just because we have measured the predictability of certain modes does not mean that this predictability can be attained; it is only an upper bound on what can be attained. Furthermore the predictability limits we report above are only valid for CCSM3. Other models and nature may well have different limits.
Beyond these unavoidable shortcomings in our investigation, there are others gaps in our study that can be addressed through additional work. Our current study has focused on prominent modes that explain 40% of the total variance. It would be worthwhile to expand the analysis from individual modes to generic forecast fields. Second, investigations in other ocean basins would be useful. In preliminary work, we have calculated the RMSD of SST and subsurface temperature in several ocean basins and find the saturation time is highly dependent on region. Figure 1b gives one example of this, namely the equatorial Pacific, where saturation is reached much earlier than in the North Pacific. Third, though we have shown that the subsurface propagating mode carries a signature in SST and atmospheric surface pressure, it would be useful to quantify the predictability of this component of lowfrequency SST and atmospheric variability. Fourth, we have not investigated the structure of those perturbations that lead to the fastest loss of predictability though such information is needed for designing the climate observational system. Fifth, we have only considered predictability of the first kind while there is a forecast range at which predictability of the second kind should dominate (Hawkins and Sutton 2009; Meehl et al. 2009). We expect to consider these topics in future investigations.
Footnotes
 1.
Note that in contrast to results for finite ensembles, here relative entropy does converge to zero.
Notes
Acknowledgment
The authors acknowledge many colleagues for useful conversations and support from the DOE under Cooperative Agreement No. DEFC0297ER62402. NCAR is sponsored by the National Science Foundation.
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