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## Abstract

Eigenvectors and eigenvalues of the nondivergent barotropic vorticity equation linearized about zonally asymmetric wintertime mean flows are calculated to determine which barotropic modes might contribute to westward propagating disturbances observed in nature. Of particular interest are modes that correspond to a recurring pattern concentrated in the Western Hemisphere with a period of about 25 days reported by Branstator and Kushnir.

The most unstable modes of November–March means from individual years tend to be westward propagating and have a structure that is similar to the observed 25-day pattern.

By following the evolution of each Rossby–Haurwitz mode as the basic state is gradually changed from a state of rest to an observed mean state, it is demonstrated that all but about eight of the Rossby–Haurwitz modes will be modified beyond recognition by the action of the time mean flow. One of these, the second gravest antisymmetric zonal wavenumber-one mode (denoted {1, 3} and sometimes referred to as the 16-day wave), has a structure that bears some resemblance to the observed 25-day pattern, but it is typically neutral. The structural similarity between this mode and the 25-day pattern is not as pronounced as the similarity between the most unstable modes and the 25-day pattern. Furthermore, the mode for the observed basic state that {1, 3) evolves to depends on the path by which the resting state is transformed into the observed state, suggesting that {1, 3} cannot always be thought of as a distinct mode in the presence of a realistic background. The results indicate that even if {1, 3) can be considered to exist in wintertime mean flows, it is distinct from the most unstable modes on those flows.

By slowly changing the basic states that support the westward propagating unstable modes until they are equal to the climatological January state that earlier studies have shown produces quasi-stationary teleconnection-like modes, it is demonstrated that the unstable westward propagating and quasi-stationary modes are related to each other.

## Abstract

Eigenvectors and eigenvalues of the nondivergent barotropic vorticity equation linearized about zonally asymmetric wintertime mean flows are calculated to determine which barotropic modes might contribute to westward propagating disturbances observed in nature. Of particular interest are modes that correspond to a recurring pattern concentrated in the Western Hemisphere with a period of about 25 days reported by Branstator and Kushnir.

The most unstable modes of November–March means from individual years tend to be westward propagating and have a structure that is similar to the observed 25-day pattern.

By following the evolution of each Rossby–Haurwitz mode as the basic state is gradually changed from a state of rest to an observed mean state, it is demonstrated that all but about eight of the Rossby–Haurwitz modes will be modified beyond recognition by the action of the time mean flow. One of these, the second gravest antisymmetric zonal wavenumber-one mode (denoted {1, 3} and sometimes referred to as the 16-day wave), has a structure that bears some resemblance to the observed 25-day pattern, but it is typically neutral. The structural similarity between this mode and the 25-day pattern is not as pronounced as the similarity between the most unstable modes and the 25-day pattern. Furthermore, the mode for the observed basic state that {1, 3) evolves to depends on the path by which the resting state is transformed into the observed state, suggesting that {1, 3} cannot always be thought of as a distinct mode in the presence of a realistic background. The results indicate that even if {1, 3) can be considered to exist in wintertime mean flows, it is distinct from the most unstable modes on those flows.

By slowly changing the basic states that support the westward propagating unstable modes until they are equal to the climatological January state that earlier studies have shown produces quasi-stationary teleconnection-like modes, it is demonstrated that the unstable westward propagating and quasi-stationary modes are related to each other.

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## Abstract

The degree to which Northern Hemisphere blocking activity is controlled by variations in zonal mean conditions is investigated.

A set of Northern Hemisphere winter season 500-hPa analyzed fields is examined for blocks using an objective index defined solely from the eddy fields. With this blocking index, it is shown that in most regions enhanced blocking activity is associated with relatively strong zonally averaged winds around 30°N and weak winds around 50°–60°N. Also, the preferred zonal positions of blocks are related to the state of the zonal mean flow. A similar analysis is carried out using data from a perpetual January GCM simulation and the same relationship between blocking activity and zonally averaged conditions is found to be valid to an even stronger degree for these data.

To investigate whether anomalous zonal mean flows are actually controlling the associated level of blocking activity, two experiments with the GCM are performed. In one experiment the zonal mean state of the GCM is forced toward a configuration that is statistically associated with enhanced blocking activity in the control simulation. In the other, the zonal mean is forced toward a state associated with suppressed blocking activity. Blocking frequency is enhanced in the first experiment and weakened in the second. Furthermore, the preferred locations for blocking in the experiments match the locations found to be associated with zonal mean anomalies in the control. This suggests the zonal mean state is influencing blocking activity.

Results from a steady barotropic linear model indicate that adjustments made by the planetary waves in reaction to anomalies in the zonal mean flow are partly responsible for the relationship between blocking and the zonal mean state.

## Abstract

The degree to which Northern Hemisphere blocking activity is controlled by variations in zonal mean conditions is investigated.

A set of Northern Hemisphere winter season 500-hPa analyzed fields is examined for blocks using an objective index defined solely from the eddy fields. With this blocking index, it is shown that in most regions enhanced blocking activity is associated with relatively strong zonally averaged winds around 30°N and weak winds around 50°–60°N. Also, the preferred zonal positions of blocks are related to the state of the zonal mean flow. A similar analysis is carried out using data from a perpetual January GCM simulation and the same relationship between blocking activity and zonally averaged conditions is found to be valid to an even stronger degree for these data.

To investigate whether anomalous zonal mean flows are actually controlling the associated level of blocking activity, two experiments with the GCM are performed. In one experiment the zonal mean state of the GCM is forced toward a configuration that is statistically associated with enhanced blocking activity in the control simulation. In the other, the zonal mean is forced toward a state associated with suppressed blocking activity. Blocking frequency is enhanced in the first experiment and weakened in the second. Furthermore, the preferred locations for blocking in the experiments match the locations found to be associated with zonal mean anomalies in the control. This suggests the zonal mean state is influencing blocking activity.

Results from a steady barotropic linear model indicate that adjustments made by the planetary waves in reaction to anomalies in the zonal mean flow are partly responsible for the relationship between blocking and the zonal mean state.

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## Abstract

This work discusses the formulation and testing of a simplified model of atmospheric dynamics. The model, which has only 200- and 700-mb streamfunctions as its prognostic fields, is designed to have a climate that approximates that of a comprehensive perpetual January general circulation model. Its governing equations are based on a Lorenz-type filtered two-layer model, but its linear terms are replaced by an empirically determined operator; the simplified model is semiempirical. Its basis consists of three-dimensional empirical orthogonal functions that are calculated using a total energy metric. The linear operator is intended to serve as a parameterization of fields, patterns, and dynamics not explicitly represented in the model. The operator is found through an optimization procedure that ensures that the semiempirical model optimally predicts streamfunction tendencies observed to occur in an extended control integration of the general circulation model.

It turns out that a model determined in this way simulates the GCM climatology quite well. The time mean state, time mean transient fluxes, and leading patterns of variability are all very similar to those in the GCM. Notable superiority over the behavior of a standard filtered two-layer model is also found. In order to understand this, calculations are undertaken to identify processes, not explicitly represented in a standard filtered two-layer model, that can be especially well parameterized linearly. Results point to a dynamical balance in the GCM such that deviations of its tendencies from the tendencies given by a standard filtered model are smaller and more nearly a linear function of streamfunction anomaly than are individual terms contributing to the deviations. An analysis of the possibility of reducing the number of basis functions in the semiempirical model shows that, whereas short-time prediction is best for the nontruncated model, in the simulation of climate mean state and transient fluxes the optimum is at rather small pattern numbers (between 30 and 70).

The leading eigenmodes of the empirically determined linear component of the simplified model are found to be nearly neutral.

## Abstract

This work discusses the formulation and testing of a simplified model of atmospheric dynamics. The model, which has only 200- and 700-mb streamfunctions as its prognostic fields, is designed to have a climate that approximates that of a comprehensive perpetual January general circulation model. Its governing equations are based on a Lorenz-type filtered two-layer model, but its linear terms are replaced by an empirically determined operator; the simplified model is semiempirical. Its basis consists of three-dimensional empirical orthogonal functions that are calculated using a total energy metric. The linear operator is intended to serve as a parameterization of fields, patterns, and dynamics not explicitly represented in the model. The operator is found through an optimization procedure that ensures that the semiempirical model optimally predicts streamfunction tendencies observed to occur in an extended control integration of the general circulation model.

It turns out that a model determined in this way simulates the GCM climatology quite well. The time mean state, time mean transient fluxes, and leading patterns of variability are all very similar to those in the GCM. Notable superiority over the behavior of a standard filtered two-layer model is also found. In order to understand this, calculations are undertaken to identify processes, not explicitly represented in a standard filtered two-layer model, that can be especially well parameterized linearly. Results point to a dynamical balance in the GCM such that deviations of its tendencies from the tendencies given by a standard filtered model are smaller and more nearly a linear function of streamfunction anomaly than are individual terms contributing to the deviations. An analysis of the possibility of reducing the number of basis functions in the semiempirical model shows that, whereas short-time prediction is best for the nontruncated model, in the simulation of climate mean state and transient fluxes the optimum is at rather small pattern numbers (between 30 and 70).

The leading eigenmodes of the empirically determined linear component of the simplified model are found to be nearly neutral.

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## Abstract

California droughts are often caused by high-amplitude and persistent ridges near and off the west coast of North America without apparent connections with ENSO. Here with a hierarchy of climate models, it is demonstrated that extreme ridges in this region are associated with a continuum of zonal wavenumber-5 circumglobal teleconnection patterns that originate from midlatitude atmospheric internal dynamics. Although tropical diabatic heating anomalies are not essential to the formation and maintenance of these wave patterns, certain persistent heating anomalies may double the probability of ridges with amplitudes in the 90th percentile occurring on interannual time scales. Those heating anomalies can be caused by either natural variability or possibly by climate change, and they do not necessarily depend on ENSO. The extreme ridges that occurred during the 2013/14 and 2014/15 winters could be examples of ridges produced by heating anomalies that are not associated with ENSO. This mechanism could provide a source of subseasonal-to-interannual predictability beyond the predictability provided by ENSO.

## Abstract

California droughts are often caused by high-amplitude and persistent ridges near and off the west coast of North America without apparent connections with ENSO. Here with a hierarchy of climate models, it is demonstrated that extreme ridges in this region are associated with a continuum of zonal wavenumber-5 circumglobal teleconnection patterns that originate from midlatitude atmospheric internal dynamics. Although tropical diabatic heating anomalies are not essential to the formation and maintenance of these wave patterns, certain persistent heating anomalies may double the probability of ridges with amplitudes in the 90th percentile occurring on interannual time scales. Those heating anomalies can be caused by either natural variability or possibly by climate change, and they do not necessarily depend on ENSO. The extreme ridges that occurred during the 2013/14 and 2014/15 winters could be examples of ridges produced by heating anomalies that are not associated with ENSO. This mechanism could provide a source of subseasonal-to-interannual predictability beyond the predictability provided by ENSO.

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## Abstract

To determine if some flow components are systematically forecast more accurately than others, 990 wintertime medium-range forecasts made at the European Centre for Medium-Range Weather Forecasts (ECMWF) are examined. It is found that forecasts skill of 500-mb extratropical large-scale heights tends to be a function of empirical orthogonal function (EOF) index, with those components that project onto the leading EOFs being markedly better forecast than the components that project onto the hailing EOFS. This is true for instantaneous forecasts of as long as 10 days’ duration. Furthermore, by answering the question, Of all possible structures which structure on average is most accurately forecast? the potential for constructing a basis that is even more adept than EOFs at distinguishing well-forecast from poorly forecast flow elements is shown. Similarly, it is found that 10-day average ECMWF forecasts, as well as 29-day average forecasts produced by a general circulation model at the National Center for Atmospheric Research, can be effectively decomposed into components that on average are either easy or difficult to predict. Using the ability to make such a decomposition, spatial filters are designed that remove those components that are usually poorly forecast. These filters can markedly improve the skill scores of medium-and extended-range forecasts, though the more effective filters substantially reduce the explained variance of the forecasts. The filters are especially effective in the extended range. For example, one filter, by removing 43% of the variance, can improve the average anomaly correlation of verified 29-day average forecasts to 0.66 from an unfiltered skill of 0.46. Such filters are proposed as a means of enhancing the utility of extended-range forecasts.

## Abstract

To determine if some flow components are systematically forecast more accurately than others, 990 wintertime medium-range forecasts made at the European Centre for Medium-Range Weather Forecasts (ECMWF) are examined. It is found that forecasts skill of 500-mb extratropical large-scale heights tends to be a function of empirical orthogonal function (EOF) index, with those components that project onto the leading EOFs being markedly better forecast than the components that project onto the hailing EOFS. This is true for instantaneous forecasts of as long as 10 days’ duration. Furthermore, by answering the question, Of all possible structures which structure on average is most accurately forecast? the potential for constructing a basis that is even more adept than EOFs at distinguishing well-forecast from poorly forecast flow elements is shown. Similarly, it is found that 10-day average ECMWF forecasts, as well as 29-day average forecasts produced by a general circulation model at the National Center for Atmospheric Research, can be effectively decomposed into components that on average are either easy or difficult to predict. Using the ability to make such a decomposition, spatial filters are designed that remove those components that are usually poorly forecast. These filters can markedly improve the skill scores of medium-and extended-range forecasts, though the more effective filters substantially reduce the explained variance of the forecasts. The filters are especially effective in the extended range. For example, one filter, by removing 43% of the variance, can improve the average anomaly correlation of verified 29-day average forecasts to 0.66 from an unfiltered skill of 0.46. Such filters are proposed as a means of enhancing the utility of extended-range forecasts.

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## Abstract

A generalization of the fluctuation–dissipation theorem (FDT) that allows generation of linear response operators that estimate the response of functionals of system state variables is tested for a system defined by an atmospheric general circulation model (AGCM). A sketch of the proof of this generalization is provided, followed by comparison of response estimates based on the theory and actual responses of the AGCM for various idealized anomalous equatorial heat sources. Tested response quantities include precipitation, variances of bandpass and low-pass streamfunction, and momentum and heat fluxes. The solutions from the FDT operators are very similar to the AGCM solutions in terms of structure while overestimating response amplitudes by about 20%. As an example of an application of such response operators, the FDT operator that estimates the response of bandpass upper-tropospheric streamfunction variance is used to find the most efficient means of disturbing the Atlantic storm tracks by tropical heating. The results of the study suggest that the generalized FDT is an attractive method for systematically studying response attributes of the climate system that are of interest to climate scientists and society.

## Abstract

A generalization of the fluctuation–dissipation theorem (FDT) that allows generation of linear response operators that estimate the response of functionals of system state variables is tested for a system defined by an atmospheric general circulation model (AGCM). A sketch of the proof of this generalization is provided, followed by comparison of response estimates based on the theory and actual responses of the AGCM for various idealized anomalous equatorial heat sources. Tested response quantities include precipitation, variances of bandpass and low-pass streamfunction, and momentum and heat fluxes. The solutions from the FDT operators are very similar to the AGCM solutions in terms of structure while overestimating response amplitudes by about 20%. As an example of an application of such response operators, the FDT operator that estimates the response of bandpass upper-tropospheric streamfunction variance is used to find the most efficient means of disturbing the Atlantic storm tracks by tropical heating. The results of the study suggest that the generalized FDT is an attractive method for systematically studying response attributes of the climate system that are of interest to climate scientists and society.

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## Abstract

The barotropic instability of time-dependent observed basic states that are periodic, with a period of 1 yr covering the complete annual cycle, is analyzed using Floquet theory. The time-dependent basic state is constructed from observed monthly averaged 300-mb streamfunction fields linearly interpolated between the different months. The propagator over the 1-yr period is constructed, and its eigenvalues and some of the fastest-growing eigenvectors, termed finite-time normal modes (FTNMs), are calculated. The fast-growing FTNMs are large-scale modes with generally largest amplitudes in the Northern Hemisphere. They exhibit intraseasonal variability in their structures, as well as longer period variations, and their amplification rates vary with time. The fastest-growing FTNM has its largest growth rate in early northern winter and its amplification has maximum cumulative effect in boreal spring when the equatorward penetration of this disturbance is also the largest. The other fast-growing FTNMs also have largest amplitudes during the first half of the year.

In all months, there are fast-growing normal modes of the monthly averaged stationary basic states that have large pattern correlations with the fastest-growing FTNM for the time-dependent basic state. For some months the individual normal modes experience dramatic local variations in growth rate; these bursts of relative growth and decay are associated with *intramodal* interference effects between the eastward and westward propagating components of a *single* traveling normal mode. Both intramodal and intermodal interference effects play significant roles in the evolution of the fastest-growing FTNM, particularly in boreal spring.

The behavior of FTNM instabilities is also examined in simplified situations including a semianalytical Floquet model in which the space and time dependencies of the stability matrix are separable. In this model, temporal variations in growth rates are directly linked to seasonality in the intensity of the climatological state.

## Abstract

The barotropic instability of time-dependent observed basic states that are periodic, with a period of 1 yr covering the complete annual cycle, is analyzed using Floquet theory. The time-dependent basic state is constructed from observed monthly averaged 300-mb streamfunction fields linearly interpolated between the different months. The propagator over the 1-yr period is constructed, and its eigenvalues and some of the fastest-growing eigenvectors, termed finite-time normal modes (FTNMs), are calculated. The fast-growing FTNMs are large-scale modes with generally largest amplitudes in the Northern Hemisphere. They exhibit intraseasonal variability in their structures, as well as longer period variations, and their amplification rates vary with time. The fastest-growing FTNM has its largest growth rate in early northern winter and its amplification has maximum cumulative effect in boreal spring when the equatorward penetration of this disturbance is also the largest. The other fast-growing FTNMs also have largest amplitudes during the first half of the year.

In all months, there are fast-growing normal modes of the monthly averaged stationary basic states that have large pattern correlations with the fastest-growing FTNM for the time-dependent basic state. For some months the individual normal modes experience dramatic local variations in growth rate; these bursts of relative growth and decay are associated with *intramodal* interference effects between the eastward and westward propagating components of a *single* traveling normal mode. Both intramodal and intermodal interference effects play significant roles in the evolution of the fastest-growing FTNM, particularly in boreal spring.

The behavior of FTNM instabilities is also examined in simplified situations including a semianalytical Floquet model in which the space and time dependencies of the stability matrix are separable. In this model, temporal variations in growth rates are directly linked to seasonality in the intensity of the climatological state.

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## Abstract

The seasonal variability of 300-hPa global streamfunction fields taken from a 40-yr period of reanalyzed observations starting on 1 January 1958 and from long 497- and 900-yr general circulation model (GCM) datasets forced by sea surface temperatures (SSTs) is examined and analyzed in terms of empirical orthogonal functions (EOFs), principal oscillation patterns (POPs), and particularly finite-time principal oscillation patterns (FTPOPs). The FTPOPs are the eigenvectors of the propagator, over a 1-yr period covering the annual cycle, that has been constructed by fitting a linear stochastic model with a time-dependent matrix operator to atmospheric fluctuations based on the daily or twice-daily 300-hPa streamfunction datasets.

The leading FTPOPs are large-scale teleconnection patterns and by construction they are the empirical analogs of finite-time normal modes (FTNMs) of linear instability theory. Hence, by comparing FTPOPs to FTNMs, the study provides insight into the ability of linear theory to explain seasonal and intraseasonal variability in the structure and growth rates of large-scale disturbances. The study finds that the leading FTPOP teleconnection patterns have similar seasonal cycles of relative growth rates and amplitudes to the leading FTNMs of the barotropic vorticity equation with 300-hPa basic states that change with the annual cycle; the largest amplitudes of both theoretical and empirical modes occur in late boreal winter or early spring, and minimum amplitudes in boreal autumn, with the GCM-based FTPOPs having additional secondary maxima in early boreal summer. In each month, there are leading POPs and EOFs that closely resemble the leading FTPOPs. Also, the growth rates of leading FTNMs and FTPOPs during each season are generally similar to those of respective leading normal modes and POPs calculated for that season. Thus the perturbations are reacting to the seasonally varying basic state faster than the state is changing and this appears to explain why linear planetary wave models with time-independent basic states can be useful. Nevertheless, *intermodal* interference effects, as well as *intramodal* interference effects, between the eastward and westward propagating components of single traveling modes, can play important roles in the evolution of FTPOPs and FTNMs, particularly in boreal spring.

This study has examined the roles of internal instability and interannual SST variability in the behavior of leading FTPOPs and has also used comparisons of FTPOPs and FTNMs for GCM simulations with and without interannually varying SSTs to assess the role of internal instability and SST variations in organizing interannual atmospheric variability. The comparison indicates that both factors are significant. The results found here also support a close relationship between the boreal spring predictability barrier of some models of climate prediction over the tropical Pacific Ocean and the amplitudes of large-scale instabilities and teleconnection patterns of the atmospheric circulation.

## Abstract

The seasonal variability of 300-hPa global streamfunction fields taken from a 40-yr period of reanalyzed observations starting on 1 January 1958 and from long 497- and 900-yr general circulation model (GCM) datasets forced by sea surface temperatures (SSTs) is examined and analyzed in terms of empirical orthogonal functions (EOFs), principal oscillation patterns (POPs), and particularly finite-time principal oscillation patterns (FTPOPs). The FTPOPs are the eigenvectors of the propagator, over a 1-yr period covering the annual cycle, that has been constructed by fitting a linear stochastic model with a time-dependent matrix operator to atmospheric fluctuations based on the daily or twice-daily 300-hPa streamfunction datasets.

The leading FTPOPs are large-scale teleconnection patterns and by construction they are the empirical analogs of finite-time normal modes (FTNMs) of linear instability theory. Hence, by comparing FTPOPs to FTNMs, the study provides insight into the ability of linear theory to explain seasonal and intraseasonal variability in the structure and growth rates of large-scale disturbances. The study finds that the leading FTPOP teleconnection patterns have similar seasonal cycles of relative growth rates and amplitudes to the leading FTNMs of the barotropic vorticity equation with 300-hPa basic states that change with the annual cycle; the largest amplitudes of both theoretical and empirical modes occur in late boreal winter or early spring, and minimum amplitudes in boreal autumn, with the GCM-based FTPOPs having additional secondary maxima in early boreal summer. In each month, there are leading POPs and EOFs that closely resemble the leading FTPOPs. Also, the growth rates of leading FTNMs and FTPOPs during each season are generally similar to those of respective leading normal modes and POPs calculated for that season. Thus the perturbations are reacting to the seasonally varying basic state faster than the state is changing and this appears to explain why linear planetary wave models with time-independent basic states can be useful. Nevertheless, *intermodal* interference effects, as well as *intramodal* interference effects, between the eastward and westward propagating components of single traveling modes, can play important roles in the evolution of FTPOPs and FTNMs, particularly in boreal spring.

This study has examined the roles of internal instability and interannual SST variability in the behavior of leading FTPOPs and has also used comparisons of FTPOPs and FTNMs for GCM simulations with and without interannually varying SSTs to assess the role of internal instability and SST variations in organizing interannual atmospheric variability. The comparison indicates that both factors are significant. The results found here also support a close relationship between the boreal spring predictability barrier of some models of climate prediction over the tropical Pacific Ocean and the amplitudes of large-scale instabilities and teleconnection patterns of the atmospheric circulation.

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## Abstract

A linear empirical model of barotropic atmospheric dynamics is constructed in which the streamfunction tendency field is optimally predicted using the concurrent streamfunction state as a predictor. The prediction equations are those resulting from performing a linear regression between tendency and state vectors. Based on the formal analogy between this model and the linear nondivergent barotropic vorticity equation, this empirical model is applied to problems normally addressed with a conventional model based on physical principles. It is found to qualitatively represent the horizontal dispersion of energy and to skillfully predict how a general circulation model will respond to steady tropical heat sources. Analysis of model solutions indicates that the empirical model’s dynamics include processes that are not represented by conventional nondivergent linear models. Most significantly, the influence of internally generated midlatitude divergence anomalies and of anomalous vorticity fluxes by high-frequency transients associated with low-frequency anomalies are automatically incorporated into the empirical model. The results suggest the utility of empirical models of atmospheric dynamics in situations where estimates of the response to external forcing are needed or as a standard of comparison in efforts to make models based on physical principles more complete.

## Abstract

A linear empirical model of barotropic atmospheric dynamics is constructed in which the streamfunction tendency field is optimally predicted using the concurrent streamfunction state as a predictor. The prediction equations are those resulting from performing a linear regression between tendency and state vectors. Based on the formal analogy between this model and the linear nondivergent barotropic vorticity equation, this empirical model is applied to problems normally addressed with a conventional model based on physical principles. It is found to qualitatively represent the horizontal dispersion of energy and to skillfully predict how a general circulation model will respond to steady tropical heat sources. Analysis of model solutions indicates that the empirical model’s dynamics include processes that are not represented by conventional nondivergent linear models. Most significantly, the influence of internally generated midlatitude divergence anomalies and of anomalous vorticity fluxes by high-frequency transients associated with low-frequency anomalies are automatically incorporated into the empirical model. The results suggest the utility of empirical models of atmospheric dynamics in situations where estimates of the response to external forcing are needed or as a standard of comparison in efforts to make models based on physical principles more complete.

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## Abstract

Using a variational procedure, we numerically search for steady solutions to the unforced, inviscid barotropic vorticity equation on the sphere. The algorithm produces many states that have extremely small tendencies within the triangular 15 spherical harmonic truncation employed in the calculation and which can thus be considered to be free modes. Often these states are similar to the planetary scale structure of the first guess fields; when observed 500 mb flow patterns are used as first guesses, the resulting free solutions can have structures reminiscent of time-mean atmospheric states. The functional relationship between streamfunction and absolute vorticity in the solutions is usually nonlinear, and thus the solutions are unlike any previously known free solutions of the barotropic vorticity equation.

Each first guess considered in the study leads to a distinct, free, steady stale, but the collection of such states is not dense in phase space. The distribution of these states is nonuniform. There seems to be a concentration of such states in the part of phase space in which the atmosphere resides. There, neighboring free states appear to be separated from each other by at most the distance that typically separates independent observed flows. Some evidence suggests that in certain cases free modes may be tightly clustered or even connected to each other.

Experiments with a forced–dissipative time–dependent model indicate that free modes like the ones we have found can influence model behavior. For sufficiently strong dissipation and forcing, the existence of such states leads to a resonant response when the forcing is chosen such that a particular free mode is close to being a solution of the forced dissipative system. Furthermore, at lower levels of dissipation, for which a free state is linearly unstable, model trajectories can still be periodically attracted to the free state. An example is given of a lime integration where the forced—dissipative system vacillates between two steady states. One of these states is a free state of the unforced inviscid barotropic vorticity equation.

We conclude that the existence of these free modes may redden the spectrum of the, atmosphere and enhance the prospects for long-range prediction.

## Abstract

Using a variational procedure, we numerically search for steady solutions to the unforced, inviscid barotropic vorticity equation on the sphere. The algorithm produces many states that have extremely small tendencies within the triangular 15 spherical harmonic truncation employed in the calculation and which can thus be considered to be free modes. Often these states are similar to the planetary scale structure of the first guess fields; when observed 500 mb flow patterns are used as first guesses, the resulting free solutions can have structures reminiscent of time-mean atmospheric states. The functional relationship between streamfunction and absolute vorticity in the solutions is usually nonlinear, and thus the solutions are unlike any previously known free solutions of the barotropic vorticity equation.

Each first guess considered in the study leads to a distinct, free, steady stale, but the collection of such states is not dense in phase space. The distribution of these states is nonuniform. There seems to be a concentration of such states in the part of phase space in which the atmosphere resides. There, neighboring free states appear to be separated from each other by at most the distance that typically separates independent observed flows. Some evidence suggests that in certain cases free modes may be tightly clustered or even connected to each other.

Experiments with a forced–dissipative time–dependent model indicate that free modes like the ones we have found can influence model behavior. For sufficiently strong dissipation and forcing, the existence of such states leads to a resonant response when the forcing is chosen such that a particular free mode is close to being a solution of the forced dissipative system. Furthermore, at lower levels of dissipation, for which a free state is linearly unstable, model trajectories can still be periodically attracted to the free state. An example is given of a lime integration where the forced—dissipative system vacillates between two steady states. One of these states is a free state of the unforced inviscid barotropic vorticity equation.

We conclude that the existence of these free modes may redden the spectrum of the, atmosphere and enhance the prospects for long-range prediction.