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## Abstract

A large class of wave structures in quasigeostrophic flow have instantaneous growth rates significantly larger than normal-mode growth rates. Since energy and potential enstrophy growth rates can be defined as functions of the perturbation structure, this structure can be varied in order to maximize the growth rates. Green's model is used as an illustration of the method. The only essential constraint in the inviscid case is the conservation of wave action. When friction is included, an appropriate modification of this constraint can be found. The problem reduces to a fourth-order nonlinear ordinary differential equation for the perturbation structure function. An analytic solution is obtained for the continuous inviscid problem with no interior potential vorticity gradient (i.e., β = 0) and yields structures that include the classical Eady wave as a special case. For the more general problem, the equation is discretized and solved numerically.A general feature of the fastest-growing baroclinic waves is that their phase tilts are more uniform with height than normal modes. As the waves evolve toward (unstable) normal-mode structures, the phase tilts concentrate at lower levels and the potential vorticity fluxes become shallower. There is no long-wave or short-wave cutoff in the initial value problem, even though the corresponding normal-mode instability may have long-wave or short-wave cutoffs or both. Indeed, the largest instantaneous growth rates in both the energy and potential enstrophy norms are obtained for the longest waves.While the variational principle requires no information about normal modes in order to find the fastest-growing structures, the principle may also be used to obtain normal modes as particular cases.

## Abstract

A large class of wave structures in quasigeostrophic flow have instantaneous growth rates significantly larger than normal-mode growth rates. Since energy and potential enstrophy growth rates can be defined as functions of the perturbation structure, this structure can be varied in order to maximize the growth rates. Green's model is used as an illustration of the method. The only essential constraint in the inviscid case is the conservation of wave action. When friction is included, an appropriate modification of this constraint can be found. The problem reduces to a fourth-order nonlinear ordinary differential equation for the perturbation structure function. An analytic solution is obtained for the continuous inviscid problem with no interior potential vorticity gradient (i.e., β = 0) and yields structures that include the classical Eady wave as a special case. For the more general problem, the equation is discretized and solved numerically.A general feature of the fastest-growing baroclinic waves is that their phase tilts are more uniform with height than normal modes. As the waves evolve toward (unstable) normal-mode structures, the phase tilts concentrate at lower levels and the potential vorticity fluxes become shallower. There is no long-wave or short-wave cutoff in the initial value problem, even though the corresponding normal-mode instability may have long-wave or short-wave cutoffs or both. Indeed, the largest instantaneous growth rates in both the energy and potential enstrophy norms are obtained for the longest waves.While the variational principle requires no information about normal modes in order to find the fastest-growing structures, the principle may also be used to obtain normal modes as particular cases.

## Abstract

A three-layer, horizontally homogeneous, quasigeostrophic model is selected as one of the simplest environments in which to study the sensitivity of baroclinic eddy fluxes in the atmosphere to the vertical structure of the basic-state temperature gradients or vertical wind shears. Eddy statistics obtained from the model are interpreted in terms of linear theory and a modified “baroclinic adjustment” hypothesis. Both linear theory and the baroclinic adjustment construction are found to provide useful predictions for the vertical structure of the eddy potential vorticity flux.For equal values of the mean vertical shear, eddy fluxes and energies are greater when the shear is concentrated at lower levels (*d*
^{2}
*U*/*dz*
^{2} < 0) than when the shear is concentrated at higher levels (*d*
^{2}
*U*/*dz*
^{2} > 0). Eddy fluxes are more sensitive to lower-than to upper-level mean temperature gradients. This relative sensitivity is a function of *γ* = *f*
^{2}Λ/(*βN*
^{2}
*H*), where Λ is the mean vertical shear and *H*is the depth of the fluid. It is enhanced as γ is reduced, as the unstable modes become shallower, until the eddies become almost completely insensitive to the strength of the upper-layer wind for *γ* < 0.5.

## Abstract

A three-layer, horizontally homogeneous, quasigeostrophic model is selected as one of the simplest environments in which to study the sensitivity of baroclinic eddy fluxes in the atmosphere to the vertical structure of the basic-state temperature gradients or vertical wind shears. Eddy statistics obtained from the model are interpreted in terms of linear theory and a modified “baroclinic adjustment” hypothesis. Both linear theory and the baroclinic adjustment construction are found to provide useful predictions for the vertical structure of the eddy potential vorticity flux.For equal values of the mean vertical shear, eddy fluxes and energies are greater when the shear is concentrated at lower levels (*d*
^{2}
*U*/*dz*
^{2} < 0) than when the shear is concentrated at higher levels (*d*
^{2}
*U*/*dz*
^{2} > 0). Eddy fluxes are more sensitive to lower-than to upper-level mean temperature gradients. This relative sensitivity is a function of *γ* = *f*
^{2}Λ/(*βN*
^{2}
*H*), where Λ is the mean vertical shear and *H*is the depth of the fluid. It is enhanced as γ is reduced, as the unstable modes become shallower, until the eddies become almost completely insensitive to the strength of the upper-layer wind for *γ* < 0.5.

## Abstract

The response of midlatitude temperature structure to changes in radiative forcing is examined in an analytical energy-balance model that includes parameterized eddy heat fluxes and linear radiative heating. The characteristics of heat-transporting baroclinic waves are determined within the model, while simultaneously allowing the waves to adjust the mean-state static stability and meridional temperature gradient. By changing the radiative equilibrium temperature, the forcing is varied through a very wide range in order to investigate qualitative, limiting behavior of the baroclinic eddy feedbacks.

The flux parameterization is based on Charney's continuously stratified, β-plane model of baroclinic instability and incorporates a parameter which is analogous to the supercriticality (i.e., degree of instability) of the Phillips' two-level model. A baroclinic adjustment hypothesis as proposed by other investigators suggests that interaction between baroclinic fluxes and radiative heating keeps the extratropical atmosphere near neutral baroclinic stability, i.e., zero supercriticality. Although the model considered here allows for this feedback, this behavior does not occur.

The model meridional temperature gradient is primarily dependent on its radiative equilibrium value and is insensitive to changes in the static stability of radiative equilibrium. The negative feedback between meridional temperature gradient and eddy heat flux is enhanced as the meridional forcing increases and the eddy flux becomes more efficient. Static stability is very sensitive to changes in the meridional forcing reflecting the strong dependence of vertical heat flux on meridional temperature gradient. A comparison of observed seasonal variation of the baroclinic stability parameter with model results suggests that a stabilizing process like moist convection is important in determining the midlatitude static and baroclinic stability during Northern Hemisphere summer.

## Abstract

The response of midlatitude temperature structure to changes in radiative forcing is examined in an analytical energy-balance model that includes parameterized eddy heat fluxes and linear radiative heating. The characteristics of heat-transporting baroclinic waves are determined within the model, while simultaneously allowing the waves to adjust the mean-state static stability and meridional temperature gradient. By changing the radiative equilibrium temperature, the forcing is varied through a very wide range in order to investigate qualitative, limiting behavior of the baroclinic eddy feedbacks.

The flux parameterization is based on Charney's continuously stratified, β-plane model of baroclinic instability and incorporates a parameter which is analogous to the supercriticality (i.e., degree of instability) of the Phillips' two-level model. A baroclinic adjustment hypothesis as proposed by other investigators suggests that interaction between baroclinic fluxes and radiative heating keeps the extratropical atmosphere near neutral baroclinic stability, i.e., zero supercriticality. Although the model considered here allows for this feedback, this behavior does not occur.

The model meridional temperature gradient is primarily dependent on its radiative equilibrium value and is insensitive to changes in the static stability of radiative equilibrium. The negative feedback between meridional temperature gradient and eddy heat flux is enhanced as the meridional forcing increases and the eddy flux becomes more efficient. Static stability is very sensitive to changes in the meridional forcing reflecting the strong dependence of vertical heat flux on meridional temperature gradient. A comparison of observed seasonal variation of the baroclinic stability parameter with model results suggests that a stabilizing process like moist convection is important in determining the midlatitude static and baroclinic stability during Northern Hemisphere summer.

## Abstract

Two conceptual paradigms have been used in the past to interpret the observed strength and structure of eddy heat fluxes in the atmosphere. One is the idea of “adjustment,” whereby the eddies respond efficiently to changes in forcing to maintain the mean isentropic slope. The other is a “diffusive” paradigm, which assumes that eddy fluxes can be parameterized in terms of the mean flow.

The relative merits of these two approaches are examined here with the aid of a two-level primitive equations model on a sphere. In most experiments the model is forced by a completely specified heating field. This eliminates the negative feedback between temperature and forcing that is present in the atmosphere and in idealized formulations such as Newtonian cooling. Thus, any intrinsic relationship that may exist between temperature gradients and the dynamical fluxes can emerge freely. As the specified forcing strength is varied, the net dynamical fluxes vary proportionately in order to maintain an equilibrium climate. There are no constraints, however, on the equilibrium level selected by the mean temperature gradients, beyond those imposed by their dynamical relationship to the fluxes.

Our experiments show that, while the dynamical fluxes adjust to the forcing quickly and efficiently to balance the heat budget, the mean temperature gradients can continue to slowly evolve. The mean meridional and vertical temperature gradients can combine in different ways to support the same eddy fluxes. For fixed forcing, the temperature gradients eventually settle to a single climate state (i.e., independent of initial conditions), but the evolution is very slow.

The model exhibits elements of both baroclinic adjustment and diffusive behavior. Adjustment operates in the sense that isentropic slopes are relatively independent of the forcing and depend only weakly on the fluxes. Diffusion works in the sense that apparently unique flux-temperature gradient relationships eventually assert themselves, however slowly. Eddy heat flux sensitivity to mean temperature gradients is in broad agreement with recent parameterization theory, given the constraints inherent in the two-level model and the structure of the forcing used in the experiments.

## Abstract

Two conceptual paradigms have been used in the past to interpret the observed strength and structure of eddy heat fluxes in the atmosphere. One is the idea of “adjustment,” whereby the eddies respond efficiently to changes in forcing to maintain the mean isentropic slope. The other is a “diffusive” paradigm, which assumes that eddy fluxes can be parameterized in terms of the mean flow.

The relative merits of these two approaches are examined here with the aid of a two-level primitive equations model on a sphere. In most experiments the model is forced by a completely specified heating field. This eliminates the negative feedback between temperature and forcing that is present in the atmosphere and in idealized formulations such as Newtonian cooling. Thus, any intrinsic relationship that may exist between temperature gradients and the dynamical fluxes can emerge freely. As the specified forcing strength is varied, the net dynamical fluxes vary proportionately in order to maintain an equilibrium climate. There are no constraints, however, on the equilibrium level selected by the mean temperature gradients, beyond those imposed by their dynamical relationship to the fluxes.

Our experiments show that, while the dynamical fluxes adjust to the forcing quickly and efficiently to balance the heat budget, the mean temperature gradients can continue to slowly evolve. The mean meridional and vertical temperature gradients can combine in different ways to support the same eddy fluxes. For fixed forcing, the temperature gradients eventually settle to a single climate state (i.e., independent of initial conditions), but the evolution is very slow.

The model exhibits elements of both baroclinic adjustment and diffusive behavior. Adjustment operates in the sense that isentropic slopes are relatively independent of the forcing and depend only weakly on the fluxes. Diffusion works in the sense that apparently unique flux-temperature gradient relationships eventually assert themselves, however slowly. Eddy heat flux sensitivity to mean temperature gradients is in broad agreement with recent parameterization theory, given the constraints inherent in the two-level model and the structure of the forcing used in the experiments.

## Abstract

The effects of topography are examined in a class of low-order quasi-geostrophic models on a midlatitude *β*-plane. In the absence of topography the models are capable of producing qualitatively realistic zonal-mean circulations. The maintenance of the zonally symmetric and asymmetric circulations are examined with different spectral truncations and topographic configurations. The response to an isolated mountain peak is the most thoroughly investigated.

When the model is run without wave–wave interactions, the time-mean wave pattern forced by the isolated mountain is a superposition of waves which are either in phase or 180° out of phase with the mountain. When they are included, transient wave-wave interactions alter the mean zonal flow, which leads to a substantial modification of the time-mean wave. Specifically, the amplitude of the longest planetary wave in the model is enhanced as that wave is pushed closer to resonance by the change in the midlevel zonal flow. A phase shift relative to the topography is also induced. A reduction in surface zonal wind caused by the nonlinear wave interactions leads to weaker topographic forcing and smaller time-mean amplitudes for shorter waves. Although the heat and vorticity budgets of the time-mean wave are dominated by “linear” wave–mean flow interactions for the planetary wave, the nonlinear advective terms are of significant magnitude and generally act to oppose the corresponding linear terms for short waves. At least five meridional modes are required to produce qualitatively realistic stationary waves, which remain relatively unchanged as the resolution is further increased.

A (5,5) model (which has 5 zonal waves, a zonal flow, and 5 meridional modes) and higher order models exhibit a significant amount of low-frequency variability and produce persistent anomalies whose time scales are not unlike those of observed anomalies. The planetary wave is not confined to a small region of phase, but undergoes considerable fluctuations in position and amplitude as evidenced by large variability in mountain-induced kinetic energy conversions. The most frequently occurring anomaly pattern can be described as an amplification and slight upstream shifting of the time-mean wave pattern. Low-frequency variability is much less pronounced in more severe truncations.

## Abstract

The effects of topography are examined in a class of low-order quasi-geostrophic models on a midlatitude *β*-plane. In the absence of topography the models are capable of producing qualitatively realistic zonal-mean circulations. The maintenance of the zonally symmetric and asymmetric circulations are examined with different spectral truncations and topographic configurations. The response to an isolated mountain peak is the most thoroughly investigated.

When the model is run without wave–wave interactions, the time-mean wave pattern forced by the isolated mountain is a superposition of waves which are either in phase or 180° out of phase with the mountain. When they are included, transient wave-wave interactions alter the mean zonal flow, which leads to a substantial modification of the time-mean wave. Specifically, the amplitude of the longest planetary wave in the model is enhanced as that wave is pushed closer to resonance by the change in the midlevel zonal flow. A phase shift relative to the topography is also induced. A reduction in surface zonal wind caused by the nonlinear wave interactions leads to weaker topographic forcing and smaller time-mean amplitudes for shorter waves. Although the heat and vorticity budgets of the time-mean wave are dominated by “linear” wave–mean flow interactions for the planetary wave, the nonlinear advective terms are of significant magnitude and generally act to oppose the corresponding linear terms for short waves. At least five meridional modes are required to produce qualitatively realistic stationary waves, which remain relatively unchanged as the resolution is further increased.

A (5,5) model (which has 5 zonal waves, a zonal flow, and 5 meridional modes) and higher order models exhibit a significant amount of low-frequency variability and produce persistent anomalies whose time scales are not unlike those of observed anomalies. The planetary wave is not confined to a small region of phase, but undergoes considerable fluctuations in position and amplitude as evidenced by large variability in mountain-induced kinetic energy conversions. The most frequently occurring anomaly pattern can be described as an amplification and slight upstream shifting of the time-mean wave pattern. Low-frequency variability is much less pronounced in more severe truncations.

## Abstract

Observational studies have revealed some coherent extratropical patterns associated with the tropical Madden–Julian (MJ) wave. This study is an attempt to clarify and constrain the interpretation of these patterns by investigating tropical–extratropical interactions on intraseasonal time scales in a global spectral model (GSM). Forcing representative of northern winter is used. A simple heating-only cumulus parameterization scheme is included to generate the MJ wave. The wave period in the model falls within the 30–60 day range observed and has a structure consistent with observations.

Various statistical techniques including compositing, empirical orthogonal function (EOF) analysis, and singular value decomposition (SVD) have been used to identify those extratropical patterns associated with the tropical MJ wave.

Under zonally symmetric external conditions (no topography) the MJ wave maintains a highly regular amplitude and phase speed. Nevertheless, there is no statistically significant coherent variability between the tropics and extratropics—no matter how much one field lags the other, and despite the frequent appearance of upper-level equatorial waveguides. Significance is determined by a Monte Carlo data scrambling method.

When topography is included, the MJ wave has a more variable amplitude in both space and time. All the statistical analyses reveal consistent planetary-scale extratropical patterns associated with different phases of the tropical wave. EOF and SVD analyses indicate that the MJ wave can explain about 10% of the variance in the extratropical 250-mb height field on intraseasonal time scales. Monte Carlo significance testing indicates that about 5% can be attributed to physical processes and 5% to chance.

The signal of tropical–extratropical interaction in the mountain case can be exposed more clearly by reducing the radiative driving by half and by using a specified propagating heat source as a proxy MJ wave. In this case up to 40% of the extratropical variance can be explained by the tropical wave, while the principal *patterns* of interaction remain similar to those obtained with strong driving.

The authors conclude that topography is essential to the propagation of a coherent MJ signal into the extratropics, while topography tends to disrupt the MJ wave in the tropics. The MJ wave (and its associated extratropical patterns) must be maintained in the presence of topography by wave activity penetrating the tropics from higher latitudes. A sufficiently high level of eddy activity in the extratropics is necessary for this to occur.

## Abstract

Observational studies have revealed some coherent extratropical patterns associated with the tropical Madden–Julian (MJ) wave. This study is an attempt to clarify and constrain the interpretation of these patterns by investigating tropical–extratropical interactions on intraseasonal time scales in a global spectral model (GSM). Forcing representative of northern winter is used. A simple heating-only cumulus parameterization scheme is included to generate the MJ wave. The wave period in the model falls within the 30–60 day range observed and has a structure consistent with observations.

Various statistical techniques including compositing, empirical orthogonal function (EOF) analysis, and singular value decomposition (SVD) have been used to identify those extratropical patterns associated with the tropical MJ wave.

Under zonally symmetric external conditions (no topography) the MJ wave maintains a highly regular amplitude and phase speed. Nevertheless, there is no statistically significant coherent variability between the tropics and extratropics—no matter how much one field lags the other, and despite the frequent appearance of upper-level equatorial waveguides. Significance is determined by a Monte Carlo data scrambling method.

When topography is included, the MJ wave has a more variable amplitude in both space and time. All the statistical analyses reveal consistent planetary-scale extratropical patterns associated with different phases of the tropical wave. EOF and SVD analyses indicate that the MJ wave can explain about 10% of the variance in the extratropical 250-mb height field on intraseasonal time scales. Monte Carlo significance testing indicates that about 5% can be attributed to physical processes and 5% to chance.

The signal of tropical–extratropical interaction in the mountain case can be exposed more clearly by reducing the radiative driving by half and by using a specified propagating heat source as a proxy MJ wave. In this case up to 40% of the extratropical variance can be explained by the tropical wave, while the principal *patterns* of interaction remain similar to those obtained with strong driving.

The authors conclude that topography is essential to the propagation of a coherent MJ signal into the extratropics, while topography tends to disrupt the MJ wave in the tropics. The MJ wave (and its associated extratropical patterns) must be maintained in the presence of topography by wave activity penetrating the tropics from higher latitudes. A sufficiently high level of eddy activity in the extratropics is necessary for this to occur.