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## Abstract

The stability of free and forced planetary waves in a β plane channel is investigated with a barotropic model. The equilibrium flows that are considered have the gravest possible scale in the meridional direction and a zonal wavenumber of either 1 or 2. The equilibrium-forced waves are the result of the interaction of a constant mean zonal wind over finite-amplitude surface orography.

The frequency of all possible small-amplitude perturbations to the equilibrium flows are calculated as a function of the strength of the mean zonal wind and of the amplitude of the orography. The forced zonal-wavenumber-1 flow is found to have three major regions of instability in parameter space, two of which have stationary growing perturbations. The free Rossby wave of that scale is stable for all amplitudes. The forced zonal-wavenumber-2 wave has two adjacent instability domains one on each side of the resonant mean zonal wind. The free wave becomes unstable for sufficiently large amplitudes. The results are interpreted through the use of a severely truncated spectral model and are related to those of previous studies with infinite β-planes. We also report the existence of a traveling subresonant topographic instability, which seems to have gone unnoticed in previous studies.

## Abstract

The stability of free and forced planetary waves in a β plane channel is investigated with a barotropic model. The equilibrium flows that are considered have the gravest possible scale in the meridional direction and a zonal wavenumber of either 1 or 2. The equilibrium-forced waves are the result of the interaction of a constant mean zonal wind over finite-amplitude surface orography.

The frequency of all possible small-amplitude perturbations to the equilibrium flows are calculated as a function of the strength of the mean zonal wind and of the amplitude of the orography. The forced zonal-wavenumber-1 flow is found to have three major regions of instability in parameter space, two of which have stationary growing perturbations. The free Rossby wave of that scale is stable for all amplitudes. The forced zonal-wavenumber-2 wave has two adjacent instability domains one on each side of the resonant mean zonal wind. The free wave becomes unstable for sufficiently large amplitudes. The results are interpreted through the use of a severely truncated spectral model and are related to those of previous studies with infinite β-planes. We also report the existence of a traveling subresonant topographic instability, which seems to have gone unnoticed in previous studies.

## Abstract

The interactions between low-frequency transients and synoptic-scale eddies are examined. The 300-mb data from 1981 to 1986 analyzed by the ECMWF are used to calculate the height tendency of the slow transients due to vorticity forcing by synoptic-scale eddies. It is found that the low-frequency transients are correlated with the forcing by the synoptic-scale eddies, most notably in the eastern part of the Pacific and Atlantic basins, where a significant portion of the time variance of the slow transient height field can be explained by the forcing. The lag correlation coefficients between the forcing and the slow transients indicate that the former leads the latter by a small phase difference (about one day).

The structure of the forcing is studied by an EOF analysis. The leading EOF mode in both oceanic sectors has a dipole structure. The pattern of the geopotential height field associated with each EOF mode of the forcing is also identified. The height pattern associated with the first Pacific forcing mode has a wave train structure, and the one associated with the first Atlantic forcing mode is a localized dipole. The correlation between the height pattern and the corresponding EOF mode in each sector is then examined, and no significant time phase shift can be found between the two.

Finally, a simple theoretical explanation is proposed to account for the phase relationship between the slow transients and their forcing by the synoptic-scale eddies.

## Abstract

The interactions between low-frequency transients and synoptic-scale eddies are examined. The 300-mb data from 1981 to 1986 analyzed by the ECMWF are used to calculate the height tendency of the slow transients due to vorticity forcing by synoptic-scale eddies. It is found that the low-frequency transients are correlated with the forcing by the synoptic-scale eddies, most notably in the eastern part of the Pacific and Atlantic basins, where a significant portion of the time variance of the slow transient height field can be explained by the forcing. The lag correlation coefficients between the forcing and the slow transients indicate that the former leads the latter by a small phase difference (about one day).

The structure of the forcing is studied by an EOF analysis. The leading EOF mode in both oceanic sectors has a dipole structure. The pattern of the geopotential height field associated with each EOF mode of the forcing is also identified. The height pattern associated with the first Pacific forcing mode has a wave train structure, and the one associated with the first Atlantic forcing mode is a localized dipole. The correlation between the height pattern and the corresponding EOF mode in each sector is then examined, and no significant time phase shift can be found between the two.

Finally, a simple theoretical explanation is proposed to account for the phase relationship between the slow transients and their forcing by the synoptic-scale eddies.

## Abstract

A primitive equations dry atmospheric model is used to investigate the atmospheric response to a tropical diabatic forcing pattern and explore how the atmospheric response changes as a function of the amplitude of the forcing. The forcing anomaly represents a linear fit of the model forcing to a tropical SST pattern of an El Niño/La Niña type. The time-averaged 500-hPa geopotential height anomaly responses of two long integrations, with forcing anomalies of equal amplitudes but opposite signs, show an asymmetric feature that is similar to observations and to previous modeling results related to El Niño and La Niña. Ensemble experiments with 61 different amplitudes of this forcing pattern are conducted. An EOF analysis of the ensemble mean of the 90-day-averaged 500-hPa height for different amplitudes of forcings shows that the leading mode of the forced variability resembles the Pacific–North American (PNA) pattern, while the second mode is a wave train across the North Atlantic to Eurasia. The relationship between the amplitude of the PNA mode and the amplitude of the forcing is linear, while the amplitude of the Atlantic/Eurasian mode has a nearly parabolic relationship with the amplitude of the forcing. A set of linear experiments with forcing perturbations and eddy flux anomalies associated with the positive and negative amplitudes of forcing conditions indicates that the nonlinearity of the extratropical response primarily results from the modification of the “basic state” caused by the large-amplitude forcing and the subsequent sensitivity of the response to that modified basic flow. A La Niña–type basic state yields a stronger response in the North Atlantic to the tropical Pacific forcing than does an El Niño–type basic state.

## Abstract

A primitive equations dry atmospheric model is used to investigate the atmospheric response to a tropical diabatic forcing pattern and explore how the atmospheric response changes as a function of the amplitude of the forcing. The forcing anomaly represents a linear fit of the model forcing to a tropical SST pattern of an El Niño/La Niña type. The time-averaged 500-hPa geopotential height anomaly responses of two long integrations, with forcing anomalies of equal amplitudes but opposite signs, show an asymmetric feature that is similar to observations and to previous modeling results related to El Niño and La Niña. Ensemble experiments with 61 different amplitudes of this forcing pattern are conducted. An EOF analysis of the ensemble mean of the 90-day-averaged 500-hPa height for different amplitudes of forcings shows that the leading mode of the forced variability resembles the Pacific–North American (PNA) pattern, while the second mode is a wave train across the North Atlantic to Eurasia. The relationship between the amplitude of the PNA mode and the amplitude of the forcing is linear, while the amplitude of the Atlantic/Eurasian mode has a nearly parabolic relationship with the amplitude of the forcing. A set of linear experiments with forcing perturbations and eddy flux anomalies associated with the positive and negative amplitudes of forcing conditions indicates that the nonlinearity of the extratropical response primarily results from the modification of the “basic state” caused by the large-amplitude forcing and the subsequent sensitivity of the response to that modified basic flow. A La Niña–type basic state yields a stronger response in the North Atlantic to the tropical Pacific forcing than does an El Niño–type basic state.

## Abstract

Three-dimensional flows for which *q*=−λ(*p*)ψ where *q* is the potential vorticity, ψ the stream function and λ some arbitrary function of pressure, are examined. It is found that flows which satisfy this condition and are quite similar to atmospheric blocking patterns can be generated by the superposition of a zonal current independent of the meridional coordinate plus two eddy components. These flows, for which the Jacobian of ψ and *q* is zero, are of interest because 1) in the absence of forcing they constitute steady state solutions of the potential vorticity equation; and 2) the possibility exists that they can be forced resonantly to a finite amplitude by means of a potential vorticity source. The arbitrariness in the choice of λ is removed by specifying the vertical profile of the diabatic heating. It is shown that when the latter is a linear function of Pressure the resultant forced flow is nearly equivalent barotropic, stable to small amplitude perturbations, with a tendency for the blocking patterns to become somewhat move prominent with increasing pressure, in rather good agreement with observations of blocking highs.

By integration of a three-level beta-plant model in time, it is shown that it is indeed possible, in the absence of dissipation, to thermally fore the above types of flows at resonance and to generate flow patterns that are quite similar to atmospheric blocking patterns. It is also shown that even when a rather broad spectrum of modes is thermally forced, the above resonant modes tend to dominate the flow, in spite of the possible interaction among modes. This would imply that provided the mean zonal flow has the proper strength to produce a resonance condition, the thermal forcing field need not have a very special structure to produce a finite amplitude disturbance through resonance.

## Abstract

Three-dimensional flows for which *q*=−λ(*p*)ψ where *q* is the potential vorticity, ψ the stream function and λ some arbitrary function of pressure, are examined. It is found that flows which satisfy this condition and are quite similar to atmospheric blocking patterns can be generated by the superposition of a zonal current independent of the meridional coordinate plus two eddy components. These flows, for which the Jacobian of ψ and *q* is zero, are of interest because 1) in the absence of forcing they constitute steady state solutions of the potential vorticity equation; and 2) the possibility exists that they can be forced resonantly to a finite amplitude by means of a potential vorticity source. The arbitrariness in the choice of λ is removed by specifying the vertical profile of the diabatic heating. It is shown that when the latter is a linear function of Pressure the resultant forced flow is nearly equivalent barotropic, stable to small amplitude perturbations, with a tendency for the blocking patterns to become somewhat move prominent with increasing pressure, in rather good agreement with observations of blocking highs.

By integration of a three-level beta-plant model in time, it is shown that it is indeed possible, in the absence of dissipation, to thermally fore the above types of flows at resonance and to generate flow patterns that are quite similar to atmospheric blocking patterns. It is also shown that even when a rather broad spectrum of modes is thermally forced, the above resonant modes tend to dominate the flow, in spite of the possible interaction among modes. This would imply that provided the mean zonal flow has the proper strength to produce a resonance condition, the thermal forcing field need not have a very special structure to produce a finite amplitude disturbance through resonance.

## Abstract

The middle-latitude standing wave problem is investigated by means of a quasi-geostrophic, linear, steady-state model in which the zonal current is perturbed by the lower boundary topography and by a distribution of heat sources and sinks. All the perturbations are assumed to have a single meridional wavelength and the dissipation is considered to take place in the surface boundary layer using, as a first approach, a horizontally uniform drag coefficient.

After investigating some basic properties of the model atmosphere, some computations are made to determine its response to the combined forcing by topography and by diabatic heating for January 1962. The resulting perturbations are found to be in rather good agreement with the observed standing waves. The results also indicate that the standing waves forced by the topography are in about the same position as those forced by the diabatic heating and that the former have somewhat larger amplitudes than the latter.

The effect of allowing the drag coefficient to have one constant value over the continents and a smaller constant value over the oceans is examined and found to be quite important when the ratio of the two values is 6, but small (yet such as to bring the computed and observed eddies into closer agreement than in the case of a uniform drag coefficient) for a ratio of 2.

## Abstract

The middle-latitude standing wave problem is investigated by means of a quasi-geostrophic, linear, steady-state model in which the zonal current is perturbed by the lower boundary topography and by a distribution of heat sources and sinks. All the perturbations are assumed to have a single meridional wavelength and the dissipation is considered to take place in the surface boundary layer using, as a first approach, a horizontally uniform drag coefficient.

After investigating some basic properties of the model atmosphere, some computations are made to determine its response to the combined forcing by topography and by diabatic heating for January 1962. The resulting perturbations are found to be in rather good agreement with the observed standing waves. The results also indicate that the standing waves forced by the topography are in about the same position as those forced by the diabatic heating and that the former have somewhat larger amplitudes than the latter.

The effect of allowing the drag coefficient to have one constant value over the continents and a smaller constant value over the oceans is examined and found to be quite important when the ratio of the two values is 6, but small (yet such as to bring the computed and observed eddies into closer agreement than in the case of a uniform drag coefficient) for a ratio of 2.

## Abstract

The resonance of stationary waves forced by topography is examined using a quasi-geostrophic model on a beta-plane channel. It is shown analytically that among the factors favoring the resonance of large, rather than synoptic or small, scale waves is the fact that the sensitivity of the large resonant responses to a change of zonal wind decreases as the scale of the resonant wave increases. A numerical model is used to examine resonance in the presence of topography having zonal wavenumber 2 with zonal flows having horizontal and vertical shear and including the effects of damping and nonlinear interactions. Although the effects of resonance are found to be important even in the presence of damping mechanisms, linear experiments with topographical forcing of reasonable amplitude indicate that a period or several weeks is required for a resonant internal mode to achieve large amplitude in the troposphere. However, as the structure of the resonant mode is such that it has much larger amplitudes in the upper atmosphere than in the troposphere, the interaction between this growing resonant mode and the mean flow which occurs when nonlinear effects are permitted triggers a stratospheric warming and zonal wind reversal. These events, which drive the system off resonance, occur long before large wave amplitudes are achieved in the lower atmosphere. The barotropic mode of zonal wavenumber 2 is shown not to resonate for reasonable values of our mean zonal wind primarily because the latter has the same (sinusoidal) meridional structure as the topography.

## Abstract

The resonance of stationary waves forced by topography is examined using a quasi-geostrophic model on a beta-plane channel. It is shown analytically that among the factors favoring the resonance of large, rather than synoptic or small, scale waves is the fact that the sensitivity of the large resonant responses to a change of zonal wind decreases as the scale of the resonant wave increases. A numerical model is used to examine resonance in the presence of topography having zonal wavenumber 2 with zonal flows having horizontal and vertical shear and including the effects of damping and nonlinear interactions. Although the effects of resonance are found to be important even in the presence of damping mechanisms, linear experiments with topographical forcing of reasonable amplitude indicate that a period or several weeks is required for a resonant internal mode to achieve large amplitude in the troposphere. However, as the structure of the resonant mode is such that it has much larger amplitudes in the upper atmosphere than in the troposphere, the interaction between this growing resonant mode and the mean flow which occurs when nonlinear effects are permitted triggers a stratospheric warming and zonal wind reversal. These events, which drive the system off resonance, occur long before large wave amplitudes are achieved in the lower atmosphere. The barotropic mode of zonal wavenumber 2 is shown not to resonate for reasonable values of our mean zonal wind primarily because the latter has the same (sinusoidal) meridional structure as the topography.

## Abstract

It is desirable to filter the unpredictable components from a medium-range forecast. Such a filtered forecast can be obtained by averaging an ensemble of predictions that started from slightly different initial atmospheric states. Different strategies have been proposed to generate the initial perturbations for such an ensemble. “Optimal” perturbation give the largest error at a prespecified forecast time. “Bred” perturbations have grown during a period prior to the analysis. “OSSE-MC” perturbations are obtained using a Monte Carlo-like observation system simulation experiment (OSSE).

In the current pilot study, the properties of the different strategies are compared. A three-level quasigeostrophic model is used to describe the evolution of the errors. The tangent linear version of this model and its adjoint version are used to generate the optimal perturbations, while bred perturbations are generated using the full nonlinear model. In the OSSE-MC method, random perturbations of model states are used in the simulation of radiosonde and satellite observations. These observations are then assimilated using an optimal interpolation (OI) assimilation system. A large OSSE-MC ensemble is obtained using such input and the OI system, which then provides the ground truth for the other ensembles. Its observed statistical properties are also used in the construction of the optimal and the bred perturbations.

The quality of the different ensemble mean medium-range forecasts is compared for forecast lengths of up to 15 days and ensembles of 2, 8, and 32 members. Before 6 days the control performs almost as well as any ensemble mean. Bred and OSSE-MC ensembles of only two members are of marginal quality. For all three methods an ensemble size of 8 is sufficient to obtain the main part of the possible improvement over the control, and all perform well for 32-member ensembles. Still better results are obtained from a weighted mean of the climate and the ensemble mean.

## Abstract

It is desirable to filter the unpredictable components from a medium-range forecast. Such a filtered forecast can be obtained by averaging an ensemble of predictions that started from slightly different initial atmospheric states. Different strategies have been proposed to generate the initial perturbations for such an ensemble. “Optimal” perturbation give the largest error at a prespecified forecast time. “Bred” perturbations have grown during a period prior to the analysis. “OSSE-MC” perturbations are obtained using a Monte Carlo-like observation system simulation experiment (OSSE).

In the current pilot study, the properties of the different strategies are compared. A three-level quasigeostrophic model is used to describe the evolution of the errors. The tangent linear version of this model and its adjoint version are used to generate the optimal perturbations, while bred perturbations are generated using the full nonlinear model. In the OSSE-MC method, random perturbations of model states are used in the simulation of radiosonde and satellite observations. These observations are then assimilated using an optimal interpolation (OI) assimilation system. A large OSSE-MC ensemble is obtained using such input and the OI system, which then provides the ground truth for the other ensembles. Its observed statistical properties are also used in the construction of the optimal and the bred perturbations.

The quality of the different ensemble mean medium-range forecasts is compared for forecast lengths of up to 15 days and ensembles of 2, 8, and 32 members. Before 6 days the control performs almost as well as any ensemble mean. Bred and OSSE-MC ensembles of only two members are of marginal quality. For all three methods an ensemble size of 8 is sufficient to obtain the main part of the possible improvement over the control, and all perform well for 32-member ensembles. Still better results are obtained from a weighted mean of the climate and the ensemble mean.

## Abstract

Numerical experiments have been performed to determine whether it is possible to improve the quality of atmospheric forecasts by using the average of two predictions starting from slightly perturbed initial conditions. The predictions are made with a T21 quasi-nondivergent three-level model and a “perfect model” approach is used, so that all prediction errors are due to the uncertainty in the initial conditions. The two perturbed predictions are initialized by adding to and subtracting from the control initial state a small-amplitude disturbance called a “bred” mode, obtained as the fastest-growing small-amplitude perturbation of the model over a 20-day period preceding the beginning of the forecast.

The results indicate that for initial states that contain very small analysis errors the two-member ensemble yields a mean forecast of lower quality than the control forecast. For larger-amplitude analysis error fields, however, the ensemble prediction outperforms the control forecast. When a statistical distribution of possible analysis errors is considered, it is found that on average the mean of the two perturbed predictions is of higher quality than the control forecast.

The study has also shown that the spread between the two perturbed predictions is correlated with the magnitude of the forecast error for every day of the forecast period from day 1 to day 10.

The same approach has been applied to Lorenz's three-component model and similar results have been obtained.

## Abstract

Numerical experiments have been performed to determine whether it is possible to improve the quality of atmospheric forecasts by using the average of two predictions starting from slightly perturbed initial conditions. The predictions are made with a T21 quasi-nondivergent three-level model and a “perfect model” approach is used, so that all prediction errors are due to the uncertainty in the initial conditions. The two perturbed predictions are initialized by adding to and subtracting from the control initial state a small-amplitude disturbance called a “bred” mode, obtained as the fastest-growing small-amplitude perturbation of the model over a 20-day period preceding the beginning of the forecast.

The results indicate that for initial states that contain very small analysis errors the two-member ensemble yields a mean forecast of lower quality than the control forecast. For larger-amplitude analysis error fields, however, the ensemble prediction outperforms the control forecast. When a statistical distribution of possible analysis errors is considered, it is found that on average the mean of the two perturbed predictions is of higher quality than the control forecast.

The study has also shown that the spread between the two perturbed predictions is correlated with the magnitude of the forecast error for every day of the forecast period from day 1 to day 10.

The same approach has been applied to Lorenz's three-component model and similar results have been obtained.

## Abstract

A 25-yr dataset is used to investigate the role of transient eddies in the dynamics of the Pacific–North American (PNA) pattern. Monthly mean vorticity and sensible heat flux divergences associated with submonthly transients are computed over the Northern Hemisphere for each winter month. These fields are composited over months with strong PNA patterns, and the average over all winter months is subtracted to obtain anomaly fields. The vorticity flux divergence anomaly is found to be well correlated with the PNA height field, particularly in the upper troposphere, where an eddy vorticity flux convergence (divergence) is found in the low (high) height regions of the PNA anomaly. The sensible heat flux divergence, on the other hand, is negatively correlated with the PNA temperature anomaly, so that the transient eddies produce a sensible heat flux out of the warm regions of the PNA and into the cold regions, thus tending to destroy the temperature anomaly.

A linear quasi-nondivergent global steady-state model is constructed using the observed climatology. The eddy vorticity and sensible heat flux divergence anomalies are treated as empirical forcing functions to simulate the response of the atmosphere. The model response to the transient-eddy forcing is found to be qualitatively similar to the PNA pattern. The amplitude of the response is weaker than observed over the North Pacific but nearly as observed over North America. The wave-activity flux computed from the model response is in reasonable agreement with that obtained from the observed PNA, except that the model shows a weaker wave activity and a spurious flux southward from the main cell of the PNA in the North Pacific. A possible explanation for this deficiency, as for the underestimation of the response in the North Pacific, is the absence of tropical forcing in the model. The results clearly show the crucial role of the transient eddies in the dynamics of the PNA, particularly over North America.

## Abstract

A 25-yr dataset is used to investigate the role of transient eddies in the dynamics of the Pacific–North American (PNA) pattern. Monthly mean vorticity and sensible heat flux divergences associated with submonthly transients are computed over the Northern Hemisphere for each winter month. These fields are composited over months with strong PNA patterns, and the average over all winter months is subtracted to obtain anomaly fields. The vorticity flux divergence anomaly is found to be well correlated with the PNA height field, particularly in the upper troposphere, where an eddy vorticity flux convergence (divergence) is found in the low (high) height regions of the PNA anomaly. The sensible heat flux divergence, on the other hand, is negatively correlated with the PNA temperature anomaly, so that the transient eddies produce a sensible heat flux out of the warm regions of the PNA and into the cold regions, thus tending to destroy the temperature anomaly.

A linear quasi-nondivergent global steady-state model is constructed using the observed climatology. The eddy vorticity and sensible heat flux divergence anomalies are treated as empirical forcing functions to simulate the response of the atmosphere. The model response to the transient-eddy forcing is found to be qualitatively similar to the PNA pattern. The amplitude of the response is weaker than observed over the North Pacific but nearly as observed over North America. The wave-activity flux computed from the model response is in reasonable agreement with that obtained from the observed PNA, except that the model shows a weaker wave activity and a spurious flux southward from the main cell of the PNA in the North Pacific. A possible explanation for this deficiency, as for the underestimation of the response in the North Pacific, is the absence of tropical forcing in the model. The results clearly show the crucial role of the transient eddies in the dynamics of the PNA, particularly over North America.

## Abstract

Based on the bivariate Madden–Julian oscillation (MJO) index defined by Wheeler and Hendon and 25 yr (1979–2004) of pentad data, the association between the North Atlantic Oscillation (NAO) and the MJO on the intraseasonal time scale during the Northern Hemisphere winter season is analyzed. Time-lagged composites and probability analysis of the NAO index for different phases of the MJO reveal a statistically significant two-way connection between the NAO and the tropical convection of the MJO. A significant increase of the NAO amplitude happens about 5–15 days after the MJO-related convection anomaly reaches the tropical Indian Ocean and western Pacific region. The development of the NAO is associated with a Rossby wave train in the upstream Pacific and North American region. In the Atlantic and African sector, there is an extratropical influence on the tropical intraseasonal variability. Certain phases of the MJO are preceded by the occurrence of strong NAOs. A significant change of upper zonal wind in the tropical Atlantic is caused by a modulated transient westerly momentum flux convergence associated with the NAO.

## Abstract

Based on the bivariate Madden–Julian oscillation (MJO) index defined by Wheeler and Hendon and 25 yr (1979–2004) of pentad data, the association between the North Atlantic Oscillation (NAO) and the MJO on the intraseasonal time scale during the Northern Hemisphere winter season is analyzed. Time-lagged composites and probability analysis of the NAO index for different phases of the MJO reveal a statistically significant two-way connection between the NAO and the tropical convection of the MJO. A significant increase of the NAO amplitude happens about 5–15 days after the MJO-related convection anomaly reaches the tropical Indian Ocean and western Pacific region. The development of the NAO is associated with a Rossby wave train in the upstream Pacific and North American region. In the Atlantic and African sector, there is an extratropical influence on the tropical intraseasonal variability. Certain phases of the MJO are preceded by the occurrence of strong NAOs. A significant change of upper zonal wind in the tropical Atlantic is caused by a modulated transient westerly momentum flux convergence associated with the NAO.