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## Abstract

Conventional and complex empirical orthogonal function (EOF) techniques show that for at least four months during the fall and winter of 1979/80 a large-amplitude, large-scale, traveling flow anomaly existed in the troposphere and stratosphere. This feature evolved in a cyclic manner, exhibiting an average period of about 23 days. Its phase propagation was toward the west, while its amplitude was concentrated in a broad envelope centered over Canada. The phenomenon was equivalent barotropic in the troposphere and increased in amplitude with altitude. It represented about 25% of the variance of geopotential heights in the troposphere and stratosphere for the months of November through March.

The meridional and zonal structure of the zonal wavenumber one component of the traveling anomaly is similar to that of the theoretical 16-day wave. However, it is distinct from that theoretical mode in that its period is longer and, in the troposphere, zonal wave two makes a large contribution to its structure.

Complex EOF analysis of many years of wintertime daily flow maps suggests that the long-lived traveling feature of 1979/80 is simply a striking example of the leading traveling pattern in the troposphere/stratosphere during the northern winter. Except for an isolated peak at about 25 days, this leading complex EOF has a red temporal spectrum.

## Abstract

Conventional and complex empirical orthogonal function (EOF) techniques show that for at least four months during the fall and winter of 1979/80 a large-amplitude, large-scale, traveling flow anomaly existed in the troposphere and stratosphere. This feature evolved in a cyclic manner, exhibiting an average period of about 23 days. Its phase propagation was toward the west, while its amplitude was concentrated in a broad envelope centered over Canada. The phenomenon was equivalent barotropic in the troposphere and increased in amplitude with altitude. It represented about 25% of the variance of geopotential heights in the troposphere and stratosphere for the months of November through March.

The meridional and zonal structure of the zonal wavenumber one component of the traveling anomaly is similar to that of the theoretical 16-day wave. However, it is distinct from that theoretical mode in that its period is longer and, in the troposphere, zonal wave two makes a large contribution to its structure.

Complex EOF analysis of many years of wintertime daily flow maps suggests that the long-lived traveling feature of 1979/80 is simply a striking example of the leading traveling pattern in the troposphere/stratosphere during the northern winter. Except for an isolated peak at about 25 days, this leading complex EOF has a red temporal spectrum.

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## Abstract

To investigate how the propagation of energy away from steady sources may be influenced by the horizontal structure of the time-averaged flow in the troposphere, solutions to a barotropic model are displayed and interpreted. The model is steady and linearized about a basic state which varies in latitude and longitude. Emphasis is placed on cases where the longitudinal variations are gradual.

Many of the results can be analyzed by locally applying tools developed for zonally symmetric basic states, e.g., ray tracing and refractive indices. Simple background flows are constructed which have longitudinally confined critical lines, reflective surfaces, wave guides, and regions conducive to propagation. An example is given in which longitudinal derivatives of the basic state are significant. The theory of ray tracing in a two-dimensional wind must be employed as an interpretive tool in this case. It is concluded that zonal variations tend to enhance poleward (equatorward) propagation in large-scale troughs (ridges).

Examples are also shown of propagation through observed January 300 mb mean flow. These experiments suggest that the southern flank of the Asiatic jet may act as a partial reflector, increasing the likelihood of resonance in the midlatitudes. The central tropical Pacific appears to have potential as a corridor for inter-hemispheric transmission, though during typical January conditions wavetrains emanating from the mid-latitudes are absorbed there.

## Abstract

To investigate how the propagation of energy away from steady sources may be influenced by the horizontal structure of the time-averaged flow in the troposphere, solutions to a barotropic model are displayed and interpreted. The model is steady and linearized about a basic state which varies in latitude and longitude. Emphasis is placed on cases where the longitudinal variations are gradual.

Many of the results can be analyzed by locally applying tools developed for zonally symmetric basic states, e.g., ray tracing and refractive indices. Simple background flows are constructed which have longitudinally confined critical lines, reflective surfaces, wave guides, and regions conducive to propagation. An example is given in which longitudinal derivatives of the basic state are significant. The theory of ray tracing in a two-dimensional wind must be employed as an interpretive tool in this case. It is concluded that zonal variations tend to enhance poleward (equatorward) propagation in large-scale troughs (ridges).

Examples are also shown of propagation through observed January 300 mb mean flow. These experiments suggest that the southern flank of the Asiatic jet may act as a partial reflector, increasing the likelihood of resonance in the midlatitudes. The central tropical Pacific appears to have potential as a corridor for inter-hemispheric transmission, though during typical January conditions wavetrains emanating from the mid-latitudes are absorbed there.

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## Abstract

The degree to which quasi-stationary midtropospheric flow is consistent with linear, potential vorticity conservation is investigated. The linear theory suggests there should be a well defined relationship between the zonal mean component of steady flow and departures from the zonal mean.

A set of 45 monthly mean winter states is examined to determine if there is a zonal/eddy relationship. Month-to-month variations in zonal mean winds are represented in terms of an EOF analysis. Correlation and compositing techniques isolate two-dimensional patterns which appear to be associated with each zonal EOF.

Solutions to the nondivergent barotropic vorticity equation linearized about a zonal mean state suggest that the two-dimensional flow patterns associated with several of the leading zonal mean EOFs are consistent with the dynamical mechanism Rossby first proposed as an explanation for a zonal/eddy relationship.

It is suggested that the zonal/eddy relationship examined here may in fact play a role in long time-scale (i.e., several weeks or longer) midtrophospheric phenomena described in previous observational studies. It may make contributions to midlatitude teleconnection patterns, tropical/extratropical relationships and, of course, the index cycle. However, the fraction of the total variance of long time-scale midtropospheric flow for which a zonal/eddy relationship holds may be small.

## Abstract

The degree to which quasi-stationary midtropospheric flow is consistent with linear, potential vorticity conservation is investigated. The linear theory suggests there should be a well defined relationship between the zonal mean component of steady flow and departures from the zonal mean.

A set of 45 monthly mean winter states is examined to determine if there is a zonal/eddy relationship. Month-to-month variations in zonal mean winds are represented in terms of an EOF analysis. Correlation and compositing techniques isolate two-dimensional patterns which appear to be associated with each zonal EOF.

Solutions to the nondivergent barotropic vorticity equation linearized about a zonal mean state suggest that the two-dimensional flow patterns associated with several of the leading zonal mean EOFs are consistent with the dynamical mechanism Rossby first proposed as an explanation for a zonal/eddy relationship.

It is suggested that the zonal/eddy relationship examined here may in fact play a role in long time-scale (i.e., several weeks or longer) midtrophospheric phenomena described in previous observational studies. It may make contributions to midlatitude teleconnection patterns, tropical/extratropical relationships and, of course, the index cycle. However, the fraction of the total variance of long time-scale midtropospheric flow for which a zonal/eddy relationship holds may be small.

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## Abstract

Experiments are presented which indicate that many features of the response of a general circulation model to sea-surface temperature anomalies in the equatorial Pacific east of the dateline can be reproduced with a linear nondivergent barotropic vorticity-conserving model. The midlatitude response to anomalous forcing is especially well reproduced by the simple model if it is linearized about the general circulation modelâ€™s wavy control climatology. Diagnosis of the linear solutions using kinetic energy and enstrophy budget, as well as indicators of group velocity, indicates that basic stateâ€“perturbation interaction supplies nearly as much energy to the perturbation flow as anomalous forcing does.

Further experiments show that the linear model is incapable of reproducing the finding of Geisler et al. that the structure of the general circulation model's midlatitude response is insensitive to the longitudinal position of the forcing anomaly. However, a Greenâ€™s function analysis of the linear model points out that the midlatitude pattern which dominates the general circulation model experiments is very easily forced by anomalies over the East Indies. Thus it may be that anomalous precipitation in that region, caused by a weakening of the Walker circulation, is the primary impetus for the midlatitude flow anomalies.

## Abstract

Experiments are presented which indicate that many features of the response of a general circulation model to sea-surface temperature anomalies in the equatorial Pacific east of the dateline can be reproduced with a linear nondivergent barotropic vorticity-conserving model. The midlatitude response to anomalous forcing is especially well reproduced by the simple model if it is linearized about the general circulation modelâ€™s wavy control climatology. Diagnosis of the linear solutions using kinetic energy and enstrophy budget, as well as indicators of group velocity, indicates that basic stateâ€“perturbation interaction supplies nearly as much energy to the perturbation flow as anomalous forcing does.

Further experiments show that the linear model is incapable of reproducing the finding of Geisler et al. that the structure of the general circulation model's midlatitude response is insensitive to the longitudinal position of the forcing anomaly. However, a Greenâ€™s function analysis of the linear model points out that the midlatitude pattern which dominates the general circulation model experiments is very easily forced by anomalies over the East Indies. Thus it may be that anomalous precipitation in that region, caused by a weakening of the Walker circulation, is the primary impetus for the midlatitude flow anomalies.

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## Abstract

An eigenanalysis of the barotropic vorticity equation linearized about 300 mb climatological flow from a general circulation model (the NCAR CCM) control simulation is described. The goal is to determine if the known behavior of the Community Climate Model (CCM) during equatorial Pacific sea-surface temperature anomaly experiments can be interpreted in terms of the linear modes calculated in the analysis.

Given that modes which would be expected to dominate steadily forced flow would either have rapid growth rates or neutral growth rates and long periods, inspection of the spectrum that results from the eigenanalysis suggests no one mode is likely to be markedly more important than all others. However, the structures of some of the leading modes are similar to anomalies produced by sea-surface temperatures in the CCM. This is especially true of the most rapidly growing mode, which is also more sensitive to steady forcing than any other mode. The (undamped) *e*-folding time of the fastest growing mode, 15 days, is rather lengthy.

Using the eigenfunctions of the adjoint system, some of the forced linear solutions, which Part I of this paper showed to resemble CCM solutions, are expanded in the eigenbasis. Those modes with long periods control these solutions but no one mode dominates. The structure of the adjoint eigenfunction which corresponds to the fastest growing eigenfunction has most of its amplitude in southern Asia and the North Indian Ocean.

The fastest growing mode, which relies more on meridional than zonal gradients for its energy source, is shown to be rather insensitive to details of the model formulation.

## Abstract

An eigenanalysis of the barotropic vorticity equation linearized about 300 mb climatological flow from a general circulation model (the NCAR CCM) control simulation is described. The goal is to determine if the known behavior of the Community Climate Model (CCM) during equatorial Pacific sea-surface temperature anomaly experiments can be interpreted in terms of the linear modes calculated in the analysis.

Given that modes which would be expected to dominate steadily forced flow would either have rapid growth rates or neutral growth rates and long periods, inspection of the spectrum that results from the eigenanalysis suggests no one mode is likely to be markedly more important than all others. However, the structures of some of the leading modes are similar to anomalies produced by sea-surface temperatures in the CCM. This is especially true of the most rapidly growing mode, which is also more sensitive to steady forcing than any other mode. The (undamped) *e*-folding time of the fastest growing mode, 15 days, is rather lengthy.

Using the eigenfunctions of the adjoint system, some of the forced linear solutions, which Part I of this paper showed to resemble CCM solutions, are expanded in the eigenbasis. Those modes with long periods control these solutions but no one mode dominates. The structure of the adjoint eigenfunction which corresponds to the fastest growing eigenfunction has most of its amplitude in southern Asia and the North Indian Ocean.

The fastest growing mode, which relies more on meridional than zonal gradients for its energy source, is shown to be rather insensitive to details of the model formulation.

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## Abstract

A linear nine-level primitive equation model is used to determine whether the nonuniform geographical distribution of low-frequency variability and the underlying structure of low-frequency patterns can be attributed to inhomogeneities in the time-mean state of the atmosphere. The linear model's behavior is compared with the low-frequency behavior of a long simulation that has been performed with a general circulation model (GCM) whose numerical formulation matches that of the linear model.

The variance of upper tropospheric streamfunction among 1000 steady linear responses to random distributions of thermal forcing is shown to be maximized in the same locations (particularly the North Pacific) as the low-frequency variance in the GCM. The only mechanisms available for such localization of variance in the linear model result from the model's basic state being a function of all three space dimensions. Empirical orthogonal function analysis of horizontal structure indicates that several recurring low-frequency patterns in the thermally forced linear model are quite similar to some of the GCM's principal monthly mean patterns. Furthermore, these recurring patterns in the linear model, like their GCM counterparts, have a distinct preference for external vertical structures. Random vorticity sources are also found to be a means of exciting one of the leading GCM patterns, while random divergence sources are most adept at stimulating very large-scale tropical circulations with upper and lower tropospheric features that are of opposite sign. Not all of the GCM's principle low-frequency patterns are found to be preferred structures of the linear model.

An energetics analysis of some of the principal low-frequency linear patterns shows that their primary source of energy is the conversion of basic state kinetic energy to perturbation kinetic energy. This is the same source that has been found to be of importance in barotropic models with zonally varying backgrounds.

## Abstract

A linear nine-level primitive equation model is used to determine whether the nonuniform geographical distribution of low-frequency variability and the underlying structure of low-frequency patterns can be attributed to inhomogeneities in the time-mean state of the atmosphere. The linear model's behavior is compared with the low-frequency behavior of a long simulation that has been performed with a general circulation model (GCM) whose numerical formulation matches that of the linear model.

The variance of upper tropospheric streamfunction among 1000 steady linear responses to random distributions of thermal forcing is shown to be maximized in the same locations (particularly the North Pacific) as the low-frequency variance in the GCM. The only mechanisms available for such localization of variance in the linear model result from the model's basic state being a function of all three space dimensions. Empirical orthogonal function analysis of horizontal structure indicates that several recurring low-frequency patterns in the thermally forced linear model are quite similar to some of the GCM's principal monthly mean patterns. Furthermore, these recurring patterns in the linear model, like their GCM counterparts, have a distinct preference for external vertical structures. Random vorticity sources are also found to be a means of exciting one of the leading GCM patterns, while random divergence sources are most adept at stimulating very large-scale tropical circulations with upper and lower tropospheric features that are of opposite sign. Not all of the GCM's principle low-frequency patterns are found to be preferred structures of the linear model.

An energetics analysis of some of the principal low-frequency linear patterns shows that their primary source of energy is the conversion of basic state kinetic energy to perturbation kinetic energy. This is the same source that has been found to be of importance in barotropic models with zonally varying backgrounds.

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## Abstract

From previous studies it is known that anomalous momentum fluxes by bandpass eddies are important in maintaining long-lasting tropospheric flow anomalies. Evidence is presented that suggests that these anomalous fluxes do not occur at random but happen because the structure of storm track activity is modified by the presence of prominent low-frequency, large-scale circulation anomalies. This behavior is noted in an extended integration of a perpetual January simulation with a general circulation model (GCM).

Because nonlinear feedbacks between low- and high-frequency variability make it difficult to establish causal relationships between these two ranges of variability, a model is constructed that approximates the storm track activity that would be expected to accompany a given low frequency without including the feedback of the high frequencies onto the low-frequency state. This model is based on the linearized primitive equations and uses a series of short integrations from random initial conditions to establish the statistics of the storm tracks. After first tuning this model to reproduce the storm tracks of the GCM climate, the model is verified by applying it to other climate states. Next it is used to determine the perturbations to the climatological storm track that result because of the presence of two prominent low-frequency anomalies that are observed to frequently occur in the GCM. The storm track anomalies that are caused by these low-frequency anomalies match the storm track anomalies that accompany the low-frequency anomalies in the GCM, thus establishing that in the GCM the low-frequency patterns cause the coincident storm track anomalies.

Further experiments with the linear storm track model help to pinpoint which processes are important in organizing the storm track structure. It turns out that the barotropic component of the low-frequency disturbances has a large impact on the high-frequency disturbances. Group velocity calculations indicate that the barotropic component affects the distribution of storms by steering, rather than stretching and straining, the perturbations. Calculations also demonstrate that because the distribution of storms in the climatological state is nonuniform, some large-scale patterns may be able to organize storm track activity in such a way that the associated momentum fluxes positively feed back onto the large-scale anomalies while other patterns cannot induce such a positive feedback.

The study concludes that since a two-way feedback has now been demonstrated between high- and low-frequency disturbances during episodes of prominent anomalies, the two timescales are inseparable and the description of recurring anomalies is incomplete without inclusion of the fast timescale fields. It is also suggested that the storm track model used in the study may serve as a useful means of parameterizing fluxes by bandpass perturbations in low-frequency models of the atmosphere.

## Abstract

From previous studies it is known that anomalous momentum fluxes by bandpass eddies are important in maintaining long-lasting tropospheric flow anomalies. Evidence is presented that suggests that these anomalous fluxes do not occur at random but happen because the structure of storm track activity is modified by the presence of prominent low-frequency, large-scale circulation anomalies. This behavior is noted in an extended integration of a perpetual January simulation with a general circulation model (GCM).

Because nonlinear feedbacks between low- and high-frequency variability make it difficult to establish causal relationships between these two ranges of variability, a model is constructed that approximates the storm track activity that would be expected to accompany a given low frequency without including the feedback of the high frequencies onto the low-frequency state. This model is based on the linearized primitive equations and uses a series of short integrations from random initial conditions to establish the statistics of the storm tracks. After first tuning this model to reproduce the storm tracks of the GCM climate, the model is verified by applying it to other climate states. Next it is used to determine the perturbations to the climatological storm track that result because of the presence of two prominent low-frequency anomalies that are observed to frequently occur in the GCM. The storm track anomalies that are caused by these low-frequency anomalies match the storm track anomalies that accompany the low-frequency anomalies in the GCM, thus establishing that in the GCM the low-frequency patterns cause the coincident storm track anomalies.

Further experiments with the linear storm track model help to pinpoint which processes are important in organizing the storm track structure. It turns out that the barotropic component of the low-frequency disturbances has a large impact on the high-frequency disturbances. Group velocity calculations indicate that the barotropic component affects the distribution of storms by steering, rather than stretching and straining, the perturbations. Calculations also demonstrate that because the distribution of storms in the climatological state is nonuniform, some large-scale patterns may be able to organize storm track activity in such a way that the associated momentum fluxes positively feed back onto the large-scale anomalies while other patterns cannot induce such a positive feedback.

The study concludes that since a two-way feedback has now been demonstrated between high- and low-frequency disturbances during episodes of prominent anomalies, the two timescales are inseparable and the description of recurring anomalies is incomplete without inclusion of the fast timescale fields. It is also suggested that the storm track model used in the study may serve as a useful means of parameterizing fluxes by bandpass perturbations in low-frequency models of the atmosphere.

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## Abstract

A linear budget is used to ascertain which of several proposed processes are most important in maintaining four prominent low-frequency perturbation patterns in a perpetual January general circulation model simulation with fixed ocean temperatures. In the budget technique the time-average model equations are separated into those terms that are in common with the steady, adiabatic linearized form of the model equations and the remaining terms, which are treated as forcing terms. By examining the linear response to each of these forcing terms for prominent episodes of the low-frequency patterns, the terms that are most important in maintaining the low-frequency patterns can be determined.The results indicate that interactions between the low-frequency patterns and the time-mean zonal asymmetries of the model climate are crucial to the maintenance of the patterns. Of equal importance are anomalous fluxes from transients, without which the low-frequency anomalies would not be maintained. Vorticity fluxes due to bandpass (1â€“7 days) fluctuations ranging over broad sectors of the globe are found to be the most important components of the maintaining flux anomalies. Of secondary importance are nonlinear interactions of the low-frequency deviations with themselves. For some patterns this interaction acts to skew the distribution of observed amplitudes. It is also found that the influence of the zonal-mean component of low-frequency perturbations on the remaining low-frequency perturbation components can be appreciable.Charts of the ability of point sources of heat and vorticity to excite the low-frequency patterns are used to interpret the budgets. These plots indicate that, in a system with time-dependent ocean temperatures, diabatic anomalies could be more instrumental in maintaining low-frequency anomalies than they are in the currently studied model. Midlatitude ocean temperature appears to be especially important in this regard. However, the collocation of regions with pronounced low-frequency anomalous vorticity fluxes from high-frequency transients and regions from which the low-frequency patterns are easily stimulated means that even if varying bottom boundary conditions are present, low-frequency maintenance by transients should continue to be important. The point source results also show that the maintaining transients are not configured optimally for forcing the low-frequency patterns. This indicates that some organizing mechanism must be affecting the maintaining transients.

## Abstract

A linear budget is used to ascertain which of several proposed processes are most important in maintaining four prominent low-frequency perturbation patterns in a perpetual January general circulation model simulation with fixed ocean temperatures. In the budget technique the time-average model equations are separated into those terms that are in common with the steady, adiabatic linearized form of the model equations and the remaining terms, which are treated as forcing terms. By examining the linear response to each of these forcing terms for prominent episodes of the low-frequency patterns, the terms that are most important in maintaining the low-frequency patterns can be determined.The results indicate that interactions between the low-frequency patterns and the time-mean zonal asymmetries of the model climate are crucial to the maintenance of the patterns. Of equal importance are anomalous fluxes from transients, without which the low-frequency anomalies would not be maintained. Vorticity fluxes due to bandpass (1â€“7 days) fluctuations ranging over broad sectors of the globe are found to be the most important components of the maintaining flux anomalies. Of secondary importance are nonlinear interactions of the low-frequency deviations with themselves. For some patterns this interaction acts to skew the distribution of observed amplitudes. It is also found that the influence of the zonal-mean component of low-frequency perturbations on the remaining low-frequency perturbation components can be appreciable.Charts of the ability of point sources of heat and vorticity to excite the low-frequency patterns are used to interpret the budgets. These plots indicate that, in a system with time-dependent ocean temperatures, diabatic anomalies could be more instrumental in maintaining low-frequency anomalies than they are in the currently studied model. Midlatitude ocean temperature appears to be especially important in this regard. However, the collocation of regions with pronounced low-frequency anomalous vorticity fluxes from high-frequency transients and regions from which the low-frequency patterns are easily stimulated means that even if varying bottom boundary conditions are present, low-frequency maintenance by transients should continue to be important. The point source results also show that the maintaining transients are not configured optimally for forcing the low-frequency patterns. This indicates that some organizing mechanism must be affecting the maintaining transients.

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## Abstract

The fluctuationâ€“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 earthâ€™s atmosphere.

As a test of the theoremâ€™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 AGCMâ€™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 fluctuationâ€“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 earthâ€™s atmosphere.

As a test of the theoremâ€™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 AGCMâ€™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 modelâ€™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 modelâ€™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.