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## Abstract

A recently developed class of semiempirical low-order models is utilized for the reexamination of several aspects of the complexity and nonlinearity of large-scale dynamics in a GCM. Given their low dimensionality, these models are quite realistic, due to the use of the primitive equations, an efficient EOF basis, and an empirical seasonally dependent linear parameterization of the impact of unresolved scales and not explicitely described processes. Fairly different results are obtained with respect to the dependence of short-term predictability or climate simulations on the number of employed degrees of freedom. Models using 500 degrees of freedom are significantly better in short-term predictions than smaller counterparts. Meaningful predictions of the first 500 EOFs are possible for 4–5 days, while the mean anomaly correlation for the leading 30 EOFs stays above 0.6 for up to 9 days. In a 30-EOF model this is only 6 days. A striking feature is found when it comes to simulations of the monthly mean states and transient fluxes: the 30-EOF model is performing just as well as the 500-EOF model. Since similar behavior is also found in the reproduction of the number and shape of the three significant cluster centroids in the January data of the GCM, one can speculate on a characteristic dimension in the range of a few tens for the large-scale part of the climate attractor. A partial failure diagnosed in the predictability of climate change by our statistical–dynamical models indicates that the employed empirical parameterizations might actually be climate dependent. Understanding their dependence on the large-scale flow could be a prerequisite for applicability to climate change studies. In a further analysis no support is found for the classic hypothesis that the observed cluster centroids, indicating multimodality in the climate statistics, can be interpreted as quasi steady states of the GCM's low-frequency dynamics.

## Abstract

A recently developed class of semiempirical low-order models is utilized for the reexamination of several aspects of the complexity and nonlinearity of large-scale dynamics in a GCM. Given their low dimensionality, these models are quite realistic, due to the use of the primitive equations, an efficient EOF basis, and an empirical seasonally dependent linear parameterization of the impact of unresolved scales and not explicitely described processes. Fairly different results are obtained with respect to the dependence of short-term predictability or climate simulations on the number of employed degrees of freedom. Models using 500 degrees of freedom are significantly better in short-term predictions than smaller counterparts. Meaningful predictions of the first 500 EOFs are possible for 4–5 days, while the mean anomaly correlation for the leading 30 EOFs stays above 0.6 for up to 9 days. In a 30-EOF model this is only 6 days. A striking feature is found when it comes to simulations of the monthly mean states and transient fluxes: the 30-EOF model is performing just as well as the 500-EOF model. Since similar behavior is also found in the reproduction of the number and shape of the three significant cluster centroids in the January data of the GCM, one can speculate on a characteristic dimension in the range of a few tens for the large-scale part of the climate attractor. A partial failure diagnosed in the predictability of climate change by our statistical–dynamical models indicates that the employed empirical parameterizations might actually be climate dependent. Understanding their dependence on the large-scale flow could be a prerequisite for applicability to climate change studies. In a further analysis no support is found for the classic hypothesis that the observed cluster centroids, indicating multimodality in the climate statistics, can be interpreted as quasi steady states of the GCM's low-frequency dynamics.

## Abstract

In a continuation of previous investigations on deterministic reduced atmosphere models with compact state space representation, two main modifications are introduced. First, primitive equation dynamics is used to describe the nonlinear interactions between resolved scales. Second, the seasonal cycle in its main aspects is incorporated. Stability considerations lead to a gridpoint formulation of the basic equations in the dynamical core. A total energy metric consistent with the equations can be derived, provided surface pressure is treated as constant in time. Using this metric, a reduction in the number of degrees of freedom is achieved by a projection onto three-dimensional empirical orthogonal functions (EOFs), each of them encompassing simultaneously all prognostic variables (winds and temperature). The impact of unresolved scales and not explicitly described physical processes is incorporated via an empirical linear parameterization. The basis patterns having been determined from 3 sigma levels from a GCM dataset, it is found that, in spite of the presence of a seasonal cycle, at most 500 are needed for describing 90% of the variance produced by the GCM. If compared to previous low-order models with quasigeostrophic dynamics, the reduced models exhibit at this and lower-order truncations, a considerably enhanced capability to predict GCM tendencies. An analysis of the dynamical impact of the empirical parameterization is given, hinting at an important role in controlling the seasonally dependent storm track dynamics.

## Abstract

In a continuation of previous investigations on deterministic reduced atmosphere models with compact state space representation, two main modifications are introduced. First, primitive equation dynamics is used to describe the nonlinear interactions between resolved scales. Second, the seasonal cycle in its main aspects is incorporated. Stability considerations lead to a gridpoint formulation of the basic equations in the dynamical core. A total energy metric consistent with the equations can be derived, provided surface pressure is treated as constant in time. Using this metric, a reduction in the number of degrees of freedom is achieved by a projection onto three-dimensional empirical orthogonal functions (EOFs), each of them encompassing simultaneously all prognostic variables (winds and temperature). The impact of unresolved scales and not explicitly described physical processes is incorporated via an empirical linear parameterization. The basis patterns having been determined from 3 sigma levels from a GCM dataset, it is found that, in spite of the presence of a seasonal cycle, at most 500 are needed for describing 90% of the variance produced by the GCM. If compared to previous low-order models with quasigeostrophic dynamics, the reduced models exhibit at this and lower-order truncations, a considerably enhanced capability to predict GCM tendencies. An analysis of the dynamical impact of the empirical parameterization is given, hinting at an important role in controlling the seasonally dependent storm track dynamics.

## 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.

## Abstract

A steady-state, linear, two-level primitive equation model is used to simulate the January standing wave pattern as a response to mountain, diabatic and transient eddy effects. The model equations are linearized around an observed zonal mean state which is a function of latitude and pressure. The mountain effect is the vertical velocity field resulting from zonal mean wind over the surface topography. The diabatic heating is calculated using parameterized forms of the heating processes. The transient-eddy effects, i.e., the flux convergence of momentum and heat by transient eddies, are computed from observations. Separate responses of the model are computed for each of the three forcing functions.

The amplitude of the response to diabatic heating is small compared to observed values. The vertical structure is highly baroclinic. At the upper level, the phase of the waves is approximately in agreement with the observations. The amplitude of the response to mountain forcing is comparable with observations. The wavelength of the response in the Pacific sector is shorter than observed. The vertical structure is equivalent barotropic. The combined response to diabatic heating and mountain forcing is dominated by the contribution from the mountains. The phase shows some agreement with the observations, but the Aleutian low is located too far to the west and an unrealistic high appears to the west of the dateline.

The amplitude of the response to transient eddy effects is comparable to the observations in middle and low latitudes. At high latitudes the amplitudes are much too large. The assumption of linearity is not valid for strong forcing at high latitudes where the zonal wind is very weak. The vertical structure of the response is almost equivalent barotropic.

A comparison of the responses to mountain and transient eddy effects shows an interesting phase relationship. The troughs produced by the transient forcing are found in the lee of the troughs produced by the mountains (very close to the ridge) indicating that transient forcing is organized by the mountain effects.

The combined model response to all three forcing functions shows good agreement with observations except at very high latitudes.

## Abstract

A steady-state, linear, two-level primitive equation model is used to simulate the January standing wave pattern as a response to mountain, diabatic and transient eddy effects. The model equations are linearized around an observed zonal mean state which is a function of latitude and pressure. The mountain effect is the vertical velocity field resulting from zonal mean wind over the surface topography. The diabatic heating is calculated using parameterized forms of the heating processes. The transient-eddy effects, i.e., the flux convergence of momentum and heat by transient eddies, are computed from observations. Separate responses of the model are computed for each of the three forcing functions.

The amplitude of the response to diabatic heating is small compared to observed values. The vertical structure is highly baroclinic. At the upper level, the phase of the waves is approximately in agreement with the observations. The amplitude of the response to mountain forcing is comparable with observations. The wavelength of the response in the Pacific sector is shorter than observed. The vertical structure is equivalent barotropic. The combined response to diabatic heating and mountain forcing is dominated by the contribution from the mountains. The phase shows some agreement with the observations, but the Aleutian low is located too far to the west and an unrealistic high appears to the west of the dateline.

The amplitude of the response to transient eddy effects is comparable to the observations in middle and low latitudes. At high latitudes the amplitudes are much too large. The assumption of linearity is not valid for strong forcing at high latitudes where the zonal wind is very weak. The vertical structure of the response is almost equivalent barotropic.

A comparison of the responses to mountain and transient eddy effects shows an interesting phase relationship. The troughs produced by the transient forcing are found in the lee of the troughs produced by the mountains (very close to the ridge) indicating that transient forcing is organized by the mountain effects.

The combined model response to all three forcing functions shows good agreement with observations except at very high latitudes.

## Abstract

The relevance of barotropic instability for the observed low-frequency variability in the atmosphere is investigated. The stability properties of the shallow-water equations on a sphere are computed for small values of Lamb's parameter (*F* = α^{2}Ω^{2}/*gH _{e}*) where

*a*is the earth's radius, Ω its angular velocity,

*g*gravity and

*H*the equivalent depth. For small values of

_{e}*F*these equations describe the horizontal structure of external and deep internal modes that are basically barotropic in the troposphere.

The stability of simple zonal flows, as well as free and forced planetary Rossby waves, has been computed as a function of *F*. This is done numerically using a hemispheric spectral model with a T13 truncation. For *F* = 0 we have tried to interpret the numerical results by analytically computing the stability properties of the flow when only one triad is considered. The results show that for increasing *F* the critical amplitudes for instability decrease slightly, but in the area of instability both growth rate and frequency of the perturbations decrease with increasing *F*. The horizontal structure of the perturbations changes only slightly. In most cases the instability process occurs within one triad which is the triad closest to resonance. An analysis in terms of unstable triads stems equally relevant for zonal and for nonzonal flows. The stability properties of the observed 400 mb Northern Hemisphere winter climatological flow show the same dependence on *F* as found for simple flow patterns: both growth rate and frequency of the perturbations decrease for increasing *F*.

## Abstract

The relevance of barotropic instability for the observed low-frequency variability in the atmosphere is investigated. The stability properties of the shallow-water equations on a sphere are computed for small values of Lamb's parameter (*F* = α^{2}Ω^{2}/*gH _{e}*) where

*a*is the earth's radius, Ω its angular velocity,

*g*gravity and

*H*the equivalent depth. For small values of

_{e}*F*these equations describe the horizontal structure of external and deep internal modes that are basically barotropic in the troposphere.

The stability of simple zonal flows, as well as free and forced planetary Rossby waves, has been computed as a function of *F*. This is done numerically using a hemispheric spectral model with a T13 truncation. For *F* = 0 we have tried to interpret the numerical results by analytically computing the stability properties of the flow when only one triad is considered. The results show that for increasing *F* the critical amplitudes for instability decrease slightly, but in the area of instability both growth rate and frequency of the perturbations decrease with increasing *F*. The horizontal structure of the perturbations changes only slightly. In most cases the instability process occurs within one triad which is the triad closest to resonance. An analysis in terms of unstable triads stems equally relevant for zonal and for nonzonal flows. The stability properties of the observed 400 mb Northern Hemisphere winter climatological flow show the same dependence on *F* as found for simple flow patterns: both growth rate and frequency of the perturbations decrease for increasing *F*.

## Abstract

We have investigated the nonlinear steady-state response of a barotropic model to an estimate of the observed anomalous tropical divergence forcing for the El Niño winter of 1982/83. The 400 mb climatological flow was made a forced solution of the model by adding a relaxation forcing. The Rayleigh friction coefficient (ε^{−1} = 20 days) was chosen such that this solution is marginally stable. The steady states were computed as a function of a dimensionless parameter α, that governs the strength of the anomalous forcing. The computed steady-state curve deviates markedly from a straight line, displaying a fold and an isolated branch. The linear steady state (α ≪ 1) compares well with the observed seasonal mean anomaly pattern. After the fold at α = 0.65, the agreement is smaller. A further increase in α after the fold results in saturation of the response. The streamfunction patterns of the isolated branch display unrealistically large amplitudes.

Time integrations show that the steady states govern the time-dependent behavior despite their unstable nature. The resulting time-mean patterns are very similar to the steady states. Periodic, quasi-periodic, and complete chaotic behavior are observed.

Increasing the Rayleigh friction coefficient to ε^{−1} = 10 days results in a disappearance of the fold as well as the isolated branch. As for ε^{−1} = 20 days, the agreement between the steady-state response and the observed pattern decreases when α is increased.

## Abstract

We have investigated the nonlinear steady-state response of a barotropic model to an estimate of the observed anomalous tropical divergence forcing for the El Niño winter of 1982/83. The 400 mb climatological flow was made a forced solution of the model by adding a relaxation forcing. The Rayleigh friction coefficient (ε^{−1} = 20 days) was chosen such that this solution is marginally stable. The steady states were computed as a function of a dimensionless parameter α, that governs the strength of the anomalous forcing. The computed steady-state curve deviates markedly from a straight line, displaying a fold and an isolated branch. The linear steady state (α ≪ 1) compares well with the observed seasonal mean anomaly pattern. After the fold at α = 0.65, the agreement is smaller. A further increase in α after the fold results in saturation of the response. The streamfunction patterns of the isolated branch display unrealistically large amplitudes.

Time integrations show that the steady states govern the time-dependent behavior despite their unstable nature. The resulting time-mean patterns are very similar to the steady states. Periodic, quasi-periodic, and complete chaotic behavior are observed.

Increasing the Rayleigh friction coefficient to ε^{−1} = 10 days results in a disappearance of the fold as well as the isolated branch. As for ε^{−1} = 20 days, the agreement between the steady-state response and the observed pattern decreases when α is increased.

## Abstract

For six consecutive seasons around the 1982–83 El Niño event the relation between observed anomalies in atmospheric circulation patterns and anomalies in various forcing mechanisms is diagnosed. A linear model is used in an attempt to simulate atmospheric anomalies as a stationary response to observed anomalies in tropical diabatic heating, mountain and transient eddy effects. The response to forcing by transient eddies is large for all seasons and is significantly correlated with observed anomalies. To what extent observed anomalous transient eddy activity is related to anomalous conditions at the earth’s surface can not be deduced from these experiments.

The midlatitude effect of tropical heating are found to be small oven during the mature phase of the 1982–83 warming event. However these results are critically dependent on the exact location of the zero wind line. The effect of the orography caused by observed anomalies in zonal mean westerly winds is small in general. In one season, when El Niño is at its maximum, the effect is comparable in magnitude to that of the transient eddies.

## Abstract

For six consecutive seasons around the 1982–83 El Niño event the relation between observed anomalies in atmospheric circulation patterns and anomalies in various forcing mechanisms is diagnosed. A linear model is used in an attempt to simulate atmospheric anomalies as a stationary response to observed anomalies in tropical diabatic heating, mountain and transient eddy effects. The response to forcing by transient eddies is large for all seasons and is significantly correlated with observed anomalies. To what extent observed anomalous transient eddy activity is related to anomalous conditions at the earth’s surface can not be deduced from these experiments.

The midlatitude effect of tropical heating are found to be small oven during the mature phase of the 1982–83 warming event. However these results are critically dependent on the exact location of the zero wind line. The effect of the orography caused by observed anomalies in zonal mean westerly winds is small in general. In one season, when El Niño is at its maximum, the effect is comparable in magnitude to that of the transient eddies.

## Abstract

A detailed investigation has been performed of the role of the different growth mechanisms (resonance, potential vorticity unshielding, and normal-mode baroclinic instability) in the evolution of optimal perturbations constructed for a two-layer Eady model and a kinetic energy norm. The two-layer Eady model is obtained by replacing the conventional upper rigid lid by a simple but realistic stratosphere. To make an unambiguous discussion possible, generally applicable techniques have been developed. At the heart of these techniques lies a description of the linear dynamics in terms of a variable number of potential vorticity building blocks (PVBs), which are zonally wavelike, vertically localized sheets of potential vorticity.

If the optimal perturbation is composed of only one PVB, the rapid surface cyclogenesis can be attributed to the growth of the surface PVB (the edge wave), which is excited by the tropospheric PVB via a linear resonance effect. If the optimal perturbation is constructed using multiple PVBs, this simple picture is modified only in the sense that PV unshielding dominates the surface amplification for a short time after initialization. The unshielding mechanism rapidly creates large streamfunction values at the surface, as a result of which the resonance effect is much stronger. A similar resonance effect between the tropospheric PVBs and the tropopause PVB acts negatively on the surface streamfunction amplification. The influence of the stratosphere to the surface development is negligible.

In all cases reported here, the growth due to traditional normal-mode baroclinic instability contributes either negative or only little to the surface development up to the optimization time of two days. It takes at least four days for the flow to become fully dominated by normal-mode growth, thereby confirming that finite-time optimal perturbation growth differs in many aspects fundamentally from asymptotic normal-mode baroclinic instability.

## Abstract

A detailed investigation has been performed of the role of the different growth mechanisms (resonance, potential vorticity unshielding, and normal-mode baroclinic instability) in the evolution of optimal perturbations constructed for a two-layer Eady model and a kinetic energy norm. The two-layer Eady model is obtained by replacing the conventional upper rigid lid by a simple but realistic stratosphere. To make an unambiguous discussion possible, generally applicable techniques have been developed. At the heart of these techniques lies a description of the linear dynamics in terms of a variable number of potential vorticity building blocks (PVBs), which are zonally wavelike, vertically localized sheets of potential vorticity.

If the optimal perturbation is composed of only one PVB, the rapid surface cyclogenesis can be attributed to the growth of the surface PVB (the edge wave), which is excited by the tropospheric PVB via a linear resonance effect. If the optimal perturbation is constructed using multiple PVBs, this simple picture is modified only in the sense that PV unshielding dominates the surface amplification for a short time after initialization. The unshielding mechanism rapidly creates large streamfunction values at the surface, as a result of which the resonance effect is much stronger. A similar resonance effect between the tropospheric PVBs and the tropopause PVB acts negatively on the surface streamfunction amplification. The influence of the stratosphere to the surface development is negligible.

In all cases reported here, the growth due to traditional normal-mode baroclinic instability contributes either negative or only little to the surface development up to the optimization time of two days. It takes at least four days for the flow to become fully dominated by normal-mode growth, thereby confirming that finite-time optimal perturbation growth differs in many aspects fundamentally from asymptotic normal-mode baroclinic instability.

## Abstract

Optimal perturbations are constructed for a two-layer *β*-plane extension of the Eady model. The surface and interior dynamics is interpreted using the concept of potential vorticity building blocks (PVBs), which are zonally wavelike, vertically confined sheets of quasigeostrophic potential vorticity. The results are compared with the Charney model and with the two-layer Eady model without *β*. The authors focus particularly on the role of the different growth mechanisms in the optimal perturbation evolution.

The optimal perturbations are constructed allowing only one PVB, three PVBs, and finally a discrete equivalent of a continuum of PVBs to be present initially. On the *f* plane only the PVB at the surface and at the tropopause can be amplified. In the presence of *β*, however, PVBs influence each other’s growth and propagation at all levels. Compared to the two-layer *f*-plane model, the inclusion of *β* slightly reduces the surface growth and propagation speed of all optimal perturbations. Responsible for the reduction are the interior PVBs, which are excited by the initial PVB after initialization. Their joint effect is almost as strong as the effect from the excited tropopause PVB, which is also negative at the surface.

If the optimal perturbation is composed of more than one PVB, the Orr mechanism dominates the initial amplification in the entire troposphere. At low levels, the interaction between the surface PVB and the interior tropospheric PVBs (in particular those near the critical level) takes over after about half a day, whereas the interaction between the tropopause PVB and the interior PVBs is responsible for the main amplification in the upper troposphere. In all cases in which more than one PVB is used, the growing normal mode configuration is not reached at optimization time.

## Abstract

Optimal perturbations are constructed for a two-layer *β*-plane extension of the Eady model. The surface and interior dynamics is interpreted using the concept of potential vorticity building blocks (PVBs), which are zonally wavelike, vertically confined sheets of quasigeostrophic potential vorticity. The results are compared with the Charney model and with the two-layer Eady model without *β*. The authors focus particularly on the role of the different growth mechanisms in the optimal perturbation evolution.

The optimal perturbations are constructed allowing only one PVB, three PVBs, and finally a discrete equivalent of a continuum of PVBs to be present initially. On the *f* plane only the PVB at the surface and at the tropopause can be amplified. In the presence of *β*, however, PVBs influence each other’s growth and propagation at all levels. Compared to the two-layer *f*-plane model, the inclusion of *β* slightly reduces the surface growth and propagation speed of all optimal perturbations. Responsible for the reduction are the interior PVBs, which are excited by the initial PVB after initialization. Their joint effect is almost as strong as the effect from the excited tropopause PVB, which is also negative at the surface.

If the optimal perturbation is composed of more than one PVB, the Orr mechanism dominates the initial amplification in the entire troposphere. At low levels, the interaction between the surface PVB and the interior tropospheric PVBs (in particular those near the critical level) takes over after about half a day, whereas the interaction between the tropopause PVB and the interior PVBs is responsible for the main amplification in the upper troposphere. In all cases in which more than one PVB is used, the growing normal mode configuration is not reached at optimization time.

## Abstract

Using a nonmodal decomposition technique based on the potential vorticity (PV) perspective, the optimal perturbation or singular vector (SV) of the Eady model without upper rigid lid is studied for a kinetic energy norm. Special emphasis is put on the role of the continuum modes (CMs) in the structure of the SV, and on the importance of resonance to the SV evolution. The basis for the SV is formed by a number of nonmodal structures, each consisting of a superposition of one CM and one edge wave, such that the initial surface potential temperature (PT) is zero. These nonmodal structures are used as PV building blocks to construct the SV. The motivation for using a nonmodal approach is that no attempt has been made so far to include the CM residing at the steering level of the surface edge wave in the perturbation, although it is known that this CM is in linear resonance with the surface edge wave.

Experiments with one PV building block in the initial disturbance show that the SV growth is dominated by the resonance effect except for small optimization times (less than 1 day), in which case the unshielding of PV and surface PT dominates the growth of the SV. The PV–PT unshielding provides additional growth to the SV and this explains the observation that the PV resides above the resonant level.

More PV building blocks are added to include PV unshielding as a third growth mechanism. Which of the three mechanisms dominates during the SV evolution depends on the region of interest (interior or surface), as well as on the optimization time and on the number of building blocks used. At the surface, resonance plays a dominant role even when a large number of building blocks is used and relatively small optimization times are used. For the interior of the domain, PV unshielding becomes the dominant growth mechanism when more than two PV building blocks are used. With increasing optimization times, the PV distribution of the SV becomes increasingly more concentrated near the steering level of the edge wave. This concentration of PV is explained by the enhanced importance of resonance for long optimization times as compared to short optimization times.

## Abstract

Using a nonmodal decomposition technique based on the potential vorticity (PV) perspective, the optimal perturbation or singular vector (SV) of the Eady model without upper rigid lid is studied for a kinetic energy norm. Special emphasis is put on the role of the continuum modes (CMs) in the structure of the SV, and on the importance of resonance to the SV evolution. The basis for the SV is formed by a number of nonmodal structures, each consisting of a superposition of one CM and one edge wave, such that the initial surface potential temperature (PT) is zero. These nonmodal structures are used as PV building blocks to construct the SV. The motivation for using a nonmodal approach is that no attempt has been made so far to include the CM residing at the steering level of the surface edge wave in the perturbation, although it is known that this CM is in linear resonance with the surface edge wave.

Experiments with one PV building block in the initial disturbance show that the SV growth is dominated by the resonance effect except for small optimization times (less than 1 day), in which case the unshielding of PV and surface PT dominates the growth of the SV. The PV–PT unshielding provides additional growth to the SV and this explains the observation that the PV resides above the resonant level.

More PV building blocks are added to include PV unshielding as a third growth mechanism. Which of the three mechanisms dominates during the SV evolution depends on the region of interest (interior or surface), as well as on the optimization time and on the number of building blocks used. At the surface, resonance plays a dominant role even when a large number of building blocks is used and relatively small optimization times are used. For the interior of the domain, PV unshielding becomes the dominant growth mechanism when more than two PV building blocks are used. With increasing optimization times, the PV distribution of the SV becomes increasingly more concentrated near the steering level of the edge wave. This concentration of PV is explained by the enhanced importance of resonance for long optimization times as compared to short optimization times.