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## Abstract

The variability in skill of NMC 72-h 500 mb forecasts during recent winter is examined. Root-mean-square error, anomaly correlation, and the Fisher *z*-transformation of the anomaly correlation are used as measures of skill. This latter score is appropriate for measuring variability because it produces nearly Gaussian distributions of scores. The annual mean skill of the forecasts improves throughout the period examined, but there is no trend in the variability of the *z*-transformed anomaly correlations. Thus model improvements do not seem to have improved forecast reliability. The temporal power spectrum of skill is red and the 1-day lag correlation of anomaly correlation scores is 0.60 during our sample period. Heights at 500 mb have a similar spectrum during the sampling period, so it may be that certain elements of the atmospheric flow can be used to predict the likely skill of a forecast. A few potential predictors are tested and some of these, e.g., the persistence of the flow and the spatial standard deviation of the predicted anomalies, are shown to predict 10 to 20% of the variance in skill.

## Abstract

The variability in skill of NMC 72-h 500 mb forecasts during recent winter is examined. Root-mean-square error, anomaly correlation, and the Fisher *z*-transformation of the anomaly correlation are used as measures of skill. This latter score is appropriate for measuring variability because it produces nearly Gaussian distributions of scores. The annual mean skill of the forecasts improves throughout the period examined, but there is no trend in the variability of the *z*-transformed anomaly correlations. Thus model improvements do not seem to have improved forecast reliability. The temporal power spectrum of skill is red and the 1-day lag correlation of anomaly correlation scores is 0.60 during our sample period. Heights at 500 mb have a similar spectrum during the sampling period, so it may be that certain elements of the atmospheric flow can be used to predict the likely skill of a forecast. A few potential predictors are tested and some of these, e.g., the persistence of the flow and the spatial standard deviation of the predicted anomalies, are shown to predict 10 to 20% of the variance in skill.

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## Abstract

The fluctuation–dissipation theorem (FDT) states that for systems with certain properties it is possible to generate a linear operator that gives the response of the system to weak external forcing simply by using covariances and lag-covariances of fluctuations of the undisturbed system. This paper points out that the theorem can be shown to hold for systems with properties very close to the properties of the earth’s atmosphere.

As a test of the theorem’s applicability to the atmosphere, a three-dimensional operator for steady responses to external forcing is constructed for data from an atmospheric general circulation model (AGCM). The response of this operator is then compared to the response of the AGCM for various heating functions. In most cases, the FDT-based operator gives three-dimensional responses that are very similar in structure and amplitude to the corresponding GCM responses. The operator is also able to give accurate estimates for the inverse problem in which one derives the forcing that will produce a given response in the AGCM. In the few cases where the operator is not accurate, it appears that the fact that the operator was constructed in a reduced space is at least partly responsible.

As an example of the potential utility of a response operator with the accuracy found here, the FDT-based operator is applied to a problem that is difficult to solve with an AGCM. It is used to generate an influence function that shows how well heating at each point on the globe excites the AGCM’s Northern Hemisphere annular mode (NAM). Most of the regions highlighted by this influence function, including the Arctic and tropical Indian Ocean, are verified by AGCM solutions as being effective locations for stimulating the NAM.

## Abstract

The fluctuation–dissipation theorem (FDT) states that for systems with certain properties it is possible to generate a linear operator that gives the response of the system to weak external forcing simply by using covariances and lag-covariances of fluctuations of the undisturbed system. This paper points out that the theorem can be shown to hold for systems with properties very close to the properties of the earth’s atmosphere.

As a test of the theorem’s applicability to the atmosphere, a three-dimensional operator for steady responses to external forcing is constructed for data from an atmospheric general circulation model (AGCM). The response of this operator is then compared to the response of the AGCM for various heating functions. In most cases, the FDT-based operator gives three-dimensional responses that are very similar in structure and amplitude to the corresponding GCM responses. The operator is also able to give accurate estimates for the inverse problem in which one derives the forcing that will produce a given response in the AGCM. In the few cases where the operator is not accurate, it appears that the fact that the operator was constructed in a reduced space is at least partly responsible.

As an example of the potential utility of a response operator with the accuracy found here, the FDT-based operator is applied to a problem that is difficult to solve with an AGCM. It is used to generate an influence function that shows how well heating at each point on the globe excites the AGCM’s Northern Hemisphere annular mode (NAM). Most of the regions highlighted by this influence function, including the Arctic and tropical Indian Ocean, are verified by AGCM solutions as being effective locations for stimulating the NAM.

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## Abstract

To identify and quantify indications of linear and nonlinear planetary wave behavior and their impact on the distribution of atmospheric states, characteristics of a very long integration of an atmospheric general circulation model (GCM) in a four-dimensional phase space are examined. The phase space is defined by the leading four empirical orthogonal functions of 500-hPa geopotential heights.

First it is established that nonlinear tendencies similar to those reported in an earlier study of the phase space behavior in this GCM have the potential to lead to non-Gaussian features in the probability density function (PDF) of planetary waves. Then using objective measures it is demonstrated that the model’s distribution of states has distinctive non-Gaussian features. These features are characterized in various subspaces of dimension as high as four. A key feature is the presence of three radial ridges of enhanced probability emanating from the mode, which is shifted away from the climatological mean. There is no evidence of multiple maxima in the full PDF, but the radial ridges lead to three distinct modes in the distribution of circulation *patterns*.

It is demonstrated that these key aspects of non-Gaussianity are captured by a two-Gaussian mixture model fitted in four dimensions. The two circulation states at the centroids of the component Gaussians are very similar to those associated with two nonlinear features identified by Branstator and Berner in their analysis of the trajectories of the GCM. These two dynamical features are locally linear, so it is concluded that the behavior of planetary waves can be conceptualized as being approximately piecewise-linear, leading to a two-Gaussian mixture with three preferred patterns.

## Abstract

To identify and quantify indications of linear and nonlinear planetary wave behavior and their impact on the distribution of atmospheric states, characteristics of a very long integration of an atmospheric general circulation model (GCM) in a four-dimensional phase space are examined. The phase space is defined by the leading four empirical orthogonal functions of 500-hPa geopotential heights.

First it is established that nonlinear tendencies similar to those reported in an earlier study of the phase space behavior in this GCM have the potential to lead to non-Gaussian features in the probability density function (PDF) of planetary waves. Then using objective measures it is demonstrated that the model’s distribution of states has distinctive non-Gaussian features. These features are characterized in various subspaces of dimension as high as four. A key feature is the presence of three radial ridges of enhanced probability emanating from the mode, which is shifted away from the climatological mean. There is no evidence of multiple maxima in the full PDF, but the radial ridges lead to three distinct modes in the distribution of circulation *patterns*.

It is demonstrated that these key aspects of non-Gaussianity are captured by a two-Gaussian mixture model fitted in four dimensions. The two circulation states at the centroids of the component Gaussians are very similar to those associated with two nonlinear features identified by Branstator and Berner in their analysis of the trajectories of the GCM. These two dynamical features are locally linear, so it is concluded that the behavior of planetary waves can be conceptualized as being approximately piecewise-linear, leading to a two-Gaussian mixture with three preferred patterns.

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## Abstract

Various aspects of the seasonal cycle of interannual variability of the observed 300-hPa streamfunction are documented and related to dynamical influences of the seasonality of the mean circulation. The stochastically excited nondivergent barotropic vorticity equation linearized about upper-tropospheric climatological mean states from each month of the year is used to identify characteristics of interannual variability that the seasonal cycle of the mean state should modulate. The result is interannual variability with (a) extratropical centers of variance that are much stronger in winter than summer and that are confined to midlatitudes during the warm season, (b) an annual cycle of preferred scales in midlatitudes with largest scales occurring during winter and a semiannual cycle of scales in the subtropics, and (c) streamfunction tendencies from interannual fluxes that adjust to the seasonally varying climatological eddies in such a way as to damp them. Because these same properties are also shown to exist in nature, it is concluded that the linear framework is a useful means of understanding the seasonality of interannual disturbances and that seasonality of the mean state leaves a pronounced imprint on interannual variability.

Analysis of an ensemble of general circulation model integrations indicates the signatures of seasonality produced in the stochastically driven linear framework are more useful for understanding intrinsic interannual variability than variability caused by seasonally varying sea surface temperature anomalies. Furthermore, it is found that the intrinsic variability of the GCM has properties very much like those in nature, another indication that organization resulting from anomalous forcing structure is not required for production of many aspects of the observed seasonality of interannual variability.

## Abstract

Various aspects of the seasonal cycle of interannual variability of the observed 300-hPa streamfunction are documented and related to dynamical influences of the seasonality of the mean circulation. The stochastically excited nondivergent barotropic vorticity equation linearized about upper-tropospheric climatological mean states from each month of the year is used to identify characteristics of interannual variability that the seasonal cycle of the mean state should modulate. The result is interannual variability with (a) extratropical centers of variance that are much stronger in winter than summer and that are confined to midlatitudes during the warm season, (b) an annual cycle of preferred scales in midlatitudes with largest scales occurring during winter and a semiannual cycle of scales in the subtropics, and (c) streamfunction tendencies from interannual fluxes that adjust to the seasonally varying climatological eddies in such a way as to damp them. Because these same properties are also shown to exist in nature, it is concluded that the linear framework is a useful means of understanding the seasonality of interannual disturbances and that seasonality of the mean state leaves a pronounced imprint on interannual variability.

Analysis of an ensemble of general circulation model integrations indicates the signatures of seasonality produced in the stochastically driven linear framework are more useful for understanding intrinsic interannual variability than variability caused by seasonally varying sea surface temperature anomalies. Furthermore, it is found that the intrinsic variability of the GCM has properties very much like those in nature, another indication that organization resulting from anomalous forcing structure is not required for production of many aspects of the observed seasonality of interannual variability.

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## Abstract

To identify and quantify indications of linear and nonlinear planetary wave behavior, characteristics of a very long integration of an atmospheric general circulation model in a four-dimensional phase space are examined. The phase space is defined by the leading four empirical orthogonal functions of 500-hPa geopotential heights, and the primary investigated characteristic is the state dependence of mean phase space tendencies. Defining the linear component of planetary wave tendencies as that part which can be captured by a least squares fit linear operator driven by additive Gaussian white noise, the study finds that there are distinct linear and nonlinear signatures. These signatures are especially easy to see in plots of mean tendencies projected onto phase space planes. For some planes the mean tendencies are highly linear, while for others there are strong departures from linearity.

The results of the analysis are found to depend strongly on the lag time used to estimate tendencies with the linear component monotonically increasing with lag time. This is shown to result from the ergodicity of the system. Using the theory of Markov models it is possible to remove the lag-dependent component of the tendencies from the results. When this is done the projected mean dynamics in some planes is found to be almost exclusively nonlinear, while in others it is nearly linear.

In the four-dimensional space the linear component of the dynamics is largely a reflection of a westward propagating Northern Hemisphere pattern concentrated over the Pacific and North America. The nonlinear signature can be approximated by two linear functions, each operating in a different region of phase space. One region is centered around a Pacific blocking pattern while the other is centered on a state with enhanced zonal symmetry. It is concluded that reduced models of the planetary waves should strive to include these state-dependent dynamics.

## Abstract

To identify and quantify indications of linear and nonlinear planetary wave behavior, characteristics of a very long integration of an atmospheric general circulation model in a four-dimensional phase space are examined. The phase space is defined by the leading four empirical orthogonal functions of 500-hPa geopotential heights, and the primary investigated characteristic is the state dependence of mean phase space tendencies. Defining the linear component of planetary wave tendencies as that part which can be captured by a least squares fit linear operator driven by additive Gaussian white noise, the study finds that there are distinct linear and nonlinear signatures. These signatures are especially easy to see in plots of mean tendencies projected onto phase space planes. For some planes the mean tendencies are highly linear, while for others there are strong departures from linearity.

The results of the analysis are found to depend strongly on the lag time used to estimate tendencies with the linear component monotonically increasing with lag time. This is shown to result from the ergodicity of the system. Using the theory of Markov models it is possible to remove the lag-dependent component of the tendencies from the results. When this is done the projected mean dynamics in some planes is found to be almost exclusively nonlinear, while in others it is nearly linear.

In the four-dimensional space the linear component of the dynamics is largely a reflection of a westward propagating Northern Hemisphere pattern concentrated over the Pacific and North America. The nonlinear signature can be approximated by two linear functions, each operating in a different region of phase space. One region is centered around a Pacific blocking pattern while the other is centered on a state with enhanced zonal symmetry. It is concluded that reduced models of the planetary waves should strive to include these state-dependent dynamics.

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## Abstract

One-point correlation maps of the subseasonal variability of 200-hPa meridional wind in nature and an atmospheric general circulation model are systematically analyzed to quantify the impact of the climatological-mean jets on tropospheric covariability as a result of the jets acting as waveguides for the propagation of Rossby waves. As anticipated by linear theory, signatures of jet influence are detected in terms of (i) the geographical position of the strongest teleconnections, (ii) the zonal orientation and extent of prominent patterns of variability, and (iii) the scale of the features that make up those patterns. Further evidence of jet waveguide influence comes from examining the seasonality of these teleconnection attributes. During winter, covariability can be essentially circumglobal, while during summer it tends to be confined within two separate sectors of the globe where the jets are especially strong. Experiments with a multilevel linear planetary wave model confirm that the analyzed characteristics of teleconnections in the waveguides can be attributed to the action of the mean state; no organization to the anomalous forcing of the atmosphere is required to produce these properties. Some attributes, however, depend on the presence of zonal variations in the climatological-mean state that are of similar scale to the teleconnection patterns themselves.

## Abstract

One-point correlation maps of the subseasonal variability of 200-hPa meridional wind in nature and an atmospheric general circulation model are systematically analyzed to quantify the impact of the climatological-mean jets on tropospheric covariability as a result of the jets acting as waveguides for the propagation of Rossby waves. As anticipated by linear theory, signatures of jet influence are detected in terms of (i) the geographical position of the strongest teleconnections, (ii) the zonal orientation and extent of prominent patterns of variability, and (iii) the scale of the features that make up those patterns. Further evidence of jet waveguide influence comes from examining the seasonality of these teleconnection attributes. During winter, covariability can be essentially circumglobal, while during summer it tends to be confined within two separate sectors of the globe where the jets are especially strong. Experiments with a multilevel linear planetary wave model confirm that the analyzed characteristics of teleconnections in the waveguides can be attributed to the action of the mean state; no organization to the anomalous forcing of the atmosphere is required to produce these properties. Some attributes, however, depend on the presence of zonal variations in the climatological-mean state that are of similar scale to the teleconnection patterns themselves.

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## Abstract

A 62-member ensemble of coupled general circulation model (GCM) simulations of the years 1940–2080, including the effects of projected greenhouse gas increases, is examined. The focus is on the interplay between the trend in the Northern Hemisphere December–February (DJF) mean state and the intrinsic modes of variability of the model atmosphere as given by the upper-tropospheric meridional wind. The structure of the leading modes and the trend are similar. Two commonly proposed explanations for this similarity are considered.

Several results suggest that this similarity in most respects is consistent with an explanation involving *patterns* that result from the model dynamics being well approximated by a linear system. Specifically, the leading intrinsic modes are similar to the leading modes of a stochastic model linearized about the mean state of the GCM atmosphere, trends in GCM tropical precipitation appear to excite the leading linear pattern, and the probability density functions (PDFs) of prominent circulation patterns are quasi-Gaussian.

There are, on the other hand, some subtle indications that an explanation for the similarity involving preferred *states* (which necessarily result from nonlinear influences) has some relevance. For example, though unimodal, PDFs of prominent patterns have departures from Gaussianity that are suggestive of a mixture of two Gaussian components. And there is some evidence of a shift in probability between the two components as the climate changes. Interestingly, contrary to the most prominent theory of the influence of nonlinearly produced preferred states on climate change, the centroids of the components also change as the climate changes. This modification of the system’s preferred states corresponds to a change in the structure of its dominant patterns. The change in pattern structure is reproduced by the linear stochastic model when its basic state is modified to correspond to the trend in the general circulation model’s mean atmospheric state. Thus, there is a two-way interaction between the trend and the modes of variability.

## Abstract

A 62-member ensemble of coupled general circulation model (GCM) simulations of the years 1940–2080, including the effects of projected greenhouse gas increases, is examined. The focus is on the interplay between the trend in the Northern Hemisphere December–February (DJF) mean state and the intrinsic modes of variability of the model atmosphere as given by the upper-tropospheric meridional wind. The structure of the leading modes and the trend are similar. Two commonly proposed explanations for this similarity are considered.

Several results suggest that this similarity in most respects is consistent with an explanation involving *patterns* that result from the model dynamics being well approximated by a linear system. Specifically, the leading intrinsic modes are similar to the leading modes of a stochastic model linearized about the mean state of the GCM atmosphere, trends in GCM tropical precipitation appear to excite the leading linear pattern, and the probability density functions (PDFs) of prominent circulation patterns are quasi-Gaussian.

There are, on the other hand, some subtle indications that an explanation for the similarity involving preferred *states* (which necessarily result from nonlinear influences) has some relevance. For example, though unimodal, PDFs of prominent patterns have departures from Gaussianity that are suggestive of a mixture of two Gaussian components. And there is some evidence of a shift in probability between the two components as the climate changes. Interestingly, contrary to the most prominent theory of the influence of nonlinearly produced preferred states on climate change, the centroids of the components also change as the climate changes. This modification of the system’s preferred states corresponds to a change in the structure of its dominant patterns. The change in pattern structure is reproduced by the linear stochastic model when its basic state is modified to correspond to the trend in the general circulation model’s mean atmospheric state. Thus, there is a two-way interaction between the trend and the modes of variability.

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## Abstract

A prominent pattern of variability of the Northern Hemisphere wintertime tropospheric planetary waves, referred to here as the Wave3 pattern, is identified from the NCEP–NCAR reanalysis. It is worthy of attention because its structure is similar to the linear trend pattern as well as the leading pattern of multidecadal variability of the planetary waves during the past half century. The Wave3 pattern is defined as the second empirical orthogonal function (EOF) of detrended December–February mean 300-hPa meridional wind *V*
_{300} and denotes a zonal shift of the ridges and troughs of the climatological flow. Although its interannual variance is roughly comparable to that of EOF1 of *V*
_{300}, which represents the Pacific–North America (PNA) pattern, its multidecadal variance is nearly twice as large as that of the PNA. Wave3 is not completely structurally or temporally distinct from the northern annular mode (NAM) but, for some attributes, the linkage of the observed trend to Wave3 is clearer than to NAM. The prominence of the Wave3 pattern is further supported by attributes of many climate models that participated in phase 3 of the Coupled Model Intercomparison Project (CMIP3). In particular, in the Community Climate System Model, version 3 (CCSM3), the Wave3 pattern is present as EOF3 of *V*
_{300} in both a fully coupled integration and a stand-alone atmospheric integration forced by climatological sea surface temperatures. Its existence in the latter experiment indicates that the pattern can be produced by atmospheric processes alone.

## Abstract

A prominent pattern of variability of the Northern Hemisphere wintertime tropospheric planetary waves, referred to here as the Wave3 pattern, is identified from the NCEP–NCAR reanalysis. It is worthy of attention because its structure is similar to the linear trend pattern as well as the leading pattern of multidecadal variability of the planetary waves during the past half century. The Wave3 pattern is defined as the second empirical orthogonal function (EOF) of detrended December–February mean 300-hPa meridional wind *V*
_{300} and denotes a zonal shift of the ridges and troughs of the climatological flow. Although its interannual variance is roughly comparable to that of EOF1 of *V*
_{300}, which represents the Pacific–North America (PNA) pattern, its multidecadal variance is nearly twice as large as that of the PNA. Wave3 is not completely structurally or temporally distinct from the northern annular mode (NAM) but, for some attributes, the linkage of the observed trend to Wave3 is clearer than to NAM. The prominence of the Wave3 pattern is further supported by attributes of many climate models that participated in phase 3 of the Coupled Model Intercomparison Project (CMIP3). In particular, in the Community Climate System Model, version 3 (CCSM3), the Wave3 pattern is present as EOF3 of *V*
_{300} in both a fully coupled integration and a stand-alone atmospheric integration forced by climatological sea surface temperatures. Its existence in the latter experiment indicates that the pattern can be produced by atmospheric processes alone.

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## Abstract

Predictability properties of the Atlantic meridional overturning circulation (AMOC) are measured and compared to those of the upper-500-m heat content in the North Atlantic based on control simulations from nine comprehensive coupled climate models. By estimating the rate at which perfect predictions from initially similar states diverge, the authors find the prediction range at which initialization loses its potential to have a positive impact on predictions. For the annual-mean AMOC, this range varies substantially from one model to another, but on average, it is about a decade. For eight of the models, this range is less than the corresponding range for heat content. For 5- and 10-yr averages, predictability is substantially greater than for annual means for both fields, but the enhancement is more for AMOC; indeed, for the averaged fields, AMOC is more predictable than heat content. Also, there are spatial patterns of AMOC that have especially high predictability. For the most predictable of these patterns, AMOC retains predictability for more than two decades in a typical model. These patterns are associated with heat content fluctuations that also have above-average predictability, which suggests that AMOC may have a positive influence on the predictability of heat content for these special structures.

## Abstract

Predictability properties of the Atlantic meridional overturning circulation (AMOC) are measured and compared to those of the upper-500-m heat content in the North Atlantic based on control simulations from nine comprehensive coupled climate models. By estimating the rate at which perfect predictions from initially similar states diverge, the authors find the prediction range at which initialization loses its potential to have a positive impact on predictions. For the annual-mean AMOC, this range varies substantially from one model to another, but on average, it is about a decade. For eight of the models, this range is less than the corresponding range for heat content. For 5- and 10-yr averages, predictability is substantially greater than for annual means for both fields, but the enhancement is more for AMOC; indeed, for the averaged fields, AMOC is more predictable than heat content. Also, there are spatial patterns of AMOC that have especially high predictability. For the most predictable of these patterns, AMOC retains predictability for more than two decades in a typical model. These patterns are associated with heat content fluctuations that also have above-average predictability, which suggests that AMOC may have a positive influence on the predictability of heat content for these special structures.

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## Abstract

When the climate system experiences time-dependent external forcing (e.g., from increases in greenhouse gas and aerosol concentrations), there are two inherent limits on the gain in skill of decadal climate predictions that can be attained from initializing with the observed ocean state. One is the classical initial-value predictability limit that is a consequence of the system being chaotic, and the other corresponds to the forecast range at which information from the initial conditions is overcome by the forced response. These limits are not caused by model errors; they correspond to limits on the range of useful forecasts that would exist even if nature behaved exactly as the model behaves. In this paper these two limits are quantified for the Community Climate System Model, version 3 (CCSM3), with several 40-member climate change scenario experiments. Predictability of the upper-300-m ocean temperature, on basin and global scales, is estimated by relative entropy from information theory. Despite some regional variations, overall, information from the ocean initial conditions exceeds that from the forced response for about 7 yr. After about a decade the classical initial-value predictability limit is reached, at which point the initial conditions have no remaining impact. Initial-value predictability receives a larger contribution from ensemble mean signals than from the distribution about the mean. Based on the two quantified limits, the conclusion is drawn that, to the extent that predictive skill relies solely on upper-ocean heat content, in CCSM3 decadal prediction beyond a range of about 10 yr is a boundary condition problem rather than an initial-value problem. Factors that the results of this study are sensitive and insensitive to are also discussed.

## Abstract

When the climate system experiences time-dependent external forcing (e.g., from increases in greenhouse gas and aerosol concentrations), there are two inherent limits on the gain in skill of decadal climate predictions that can be attained from initializing with the observed ocean state. One is the classical initial-value predictability limit that is a consequence of the system being chaotic, and the other corresponds to the forecast range at which information from the initial conditions is overcome by the forced response. These limits are not caused by model errors; they correspond to limits on the range of useful forecasts that would exist even if nature behaved exactly as the model behaves. In this paper these two limits are quantified for the Community Climate System Model, version 3 (CCSM3), with several 40-member climate change scenario experiments. Predictability of the upper-300-m ocean temperature, on basin and global scales, is estimated by relative entropy from information theory. Despite some regional variations, overall, information from the ocean initial conditions exceeds that from the forced response for about 7 yr. After about a decade the classical initial-value predictability limit is reached, at which point the initial conditions have no remaining impact. Initial-value predictability receives a larger contribution from ensemble mean signals than from the distribution about the mean. Based on the two quantified limits, the conclusion is drawn that, to the extent that predictive skill relies solely on upper-ocean heat content, in CCSM3 decadal prediction beyond a range of about 10 yr is a boundary condition problem rather than an initial-value problem. Factors that the results of this study are sensitive and insensitive to are also discussed.