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- Author or Editor: P. Malguzzi x
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Abstract
Stationary flow of a barotropic fluid in a β channel has been shown by Charney and De Vore (1979) to possess multiple-equilibrium solutions when sinusoidal topographic forcing is exerted within the region of resonance near the wavenumber of stationary Rossby waves, and nonlinear effects are taken into account. Charney and De Vore associate the different solutions with zonal and blocking states of global circulation. However, real topography is non-sinusoidal and, most of the time, observed blocking configurations display a pronounced regional character. On the other hand, the problem of superimposing different harmonics is made difficult here by the essential role played by nonlinearity in the theory of multiple equilibria.
In this paper, the mathematical problem of determining the stationary states of flow of barotropic fluid in a β plane when topography is nonsinusoidal is analyzed with the help of the perturbative assumptions that the latitudinal scale of the flow is very large and topography has the form of a slowly modulated sinusoid.
The multiple states of stationary flow described by Charney and De Vore are found to exist simultaneously in different regions of the β plane. Theoretical solutions corresponding to different kinds of resonant forcing are analyzed.
The theoretical solutions are discussed in relationship to the problem of blocking as a “regional” phenomenon and are shown to have several different features in common with observed persistent blocking patterns.
Abstract
Stationary flow of a barotropic fluid in a β channel has been shown by Charney and De Vore (1979) to possess multiple-equilibrium solutions when sinusoidal topographic forcing is exerted within the region of resonance near the wavenumber of stationary Rossby waves, and nonlinear effects are taken into account. Charney and De Vore associate the different solutions with zonal and blocking states of global circulation. However, real topography is non-sinusoidal and, most of the time, observed blocking configurations display a pronounced regional character. On the other hand, the problem of superimposing different harmonics is made difficult here by the essential role played by nonlinearity in the theory of multiple equilibria.
In this paper, the mathematical problem of determining the stationary states of flow of barotropic fluid in a β plane when topography is nonsinusoidal is analyzed with the help of the perturbative assumptions that the latitudinal scale of the flow is very large and topography has the form of a slowly modulated sinusoid.
The multiple states of stationary flow described by Charney and De Vore are found to exist simultaneously in different regions of the β plane. Theoretical solutions corresponding to different kinds of resonant forcing are analyzed.
The theoretical solutions are discussed in relationship to the problem of blocking as a “regional” phenomenon and are shown to have several different features in common with observed persistent blocking patterns.
Abstract
A semilinear (the wave-dynamics are linear with the time-evolution operator determined by the time-varying zonal flow while the zonal flow is fully nonlinear in the eddy fluxes) model of a baroclinic zonal jet is integrated, under macroscopic conditions realistic for the earth's atmosphere, for a time period of 20 years in a high resolution pseudospectral version and its asymptotic (in time) statistical properties are determined.
The model is studied as a dynamical system, both by following sequences of bifurcations from the stable. Hadley circulation and by embedding in lower dimension spaces. The model turns out to be far from amenable to weakly nonlinear approximations common in atmospheric and oceanographic literature.
The analysis of propagation of disturbances in the turbulent jet demonstrates the inadequacy of mean-field approximations usually adopted in studies of kinematics of Rossby waves, teleconnections, etc.
Abstract
A semilinear (the wave-dynamics are linear with the time-evolution operator determined by the time-varying zonal flow while the zonal flow is fully nonlinear in the eddy fluxes) model of a baroclinic zonal jet is integrated, under macroscopic conditions realistic for the earth's atmosphere, for a time period of 20 years in a high resolution pseudospectral version and its asymptotic (in time) statistical properties are determined.
The model is studied as a dynamical system, both by following sequences of bifurcations from the stable. Hadley circulation and by embedding in lower dimension spaces. The model turns out to be far from amenable to weakly nonlinear approximations common in atmospheric and oceanographic literature.
The analysis of propagation of disturbances in the turbulent jet demonstrates the inadequacy of mean-field approximations usually adopted in studies of kinematics of Rossby waves, teleconnections, etc.
Abstract
We consider the response of the barotropic vorticity equation on a zonally infinite f-plane or beta-plane to a weak localized vorticity source accompanied by weak Ekman damping. By performing an expansion about the unforced, undamped problem, we derive a solvability condition determining when a slight modification of a solution to the inviscid problem can occur as a solution to the weakly forced and damped problem. This condition states simply that forcing must balance dissipation in the average along each closed streamline. Much of the degeneracy of the inviscid problem is removed by the solvability condition. The above considerations are used to show that under a fairly general set of circumstances the weakly forced system possesses a high amplitude equilibrium state (identified with blocking) and a low amplitude equilibrium state. The high amplitude response is maintained by a local nonlinear resonance phenomenon, and requires the existence of a suitable solution to the inviscid problem, such as the “modon” solution. In contrast to cases previously discussed, the multiple equilibrium mechanism we treat is not dependent on global resonance.
Explicit examples of local multiple equilibria are constructed through numerical integrations on the f-plant and on a zonally infinite beta-channel. Through introduction of a reasonable amount of meridional confinement, a high amplitude solution can be obtained on the beta-plane without the use of a small radius of deformation.
It is suggested that transient eddy fluxes may be able to play the role of the forcing required in our model. A tentative comparison with a blocking event is presented, indicating a number of problematic aspects of the theory.
Abstract
We consider the response of the barotropic vorticity equation on a zonally infinite f-plane or beta-plane to a weak localized vorticity source accompanied by weak Ekman damping. By performing an expansion about the unforced, undamped problem, we derive a solvability condition determining when a slight modification of a solution to the inviscid problem can occur as a solution to the weakly forced and damped problem. This condition states simply that forcing must balance dissipation in the average along each closed streamline. Much of the degeneracy of the inviscid problem is removed by the solvability condition. The above considerations are used to show that under a fairly general set of circumstances the weakly forced system possesses a high amplitude equilibrium state (identified with blocking) and a low amplitude equilibrium state. The high amplitude response is maintained by a local nonlinear resonance phenomenon, and requires the existence of a suitable solution to the inviscid problem, such as the “modon” solution. In contrast to cases previously discussed, the multiple equilibrium mechanism we treat is not dependent on global resonance.
Explicit examples of local multiple equilibria are constructed through numerical integrations on the f-plant and on a zonally infinite beta-channel. Through introduction of a reasonable amount of meridional confinement, a high amplitude solution can be obtained on the beta-plane without the use of a small radius of deformation.
It is suggested that transient eddy fluxes may be able to play the role of the forcing required in our model. A tentative comparison with a blocking event is presented, indicating a number of problematic aspects of the theory.
Abstract
This work is an extension of the normal mode theory of lee cyclogenesis in that it removes all the simplifications and restrictive assumptions contained in previous formulations Linearized primitive equations in isentropic coordinates are integrated in time to find the most unstable eigenmode. The appropriate boundary condition for such a model applied either at z = 0 or z = h is derived. Results obtained in the β-plane with a realistically steep and high isolated topography are discussed in relation to the basic state characteristics. Limits and prospects with respect to theory verification are also discussed.
Abstract
This work is an extension of the normal mode theory of lee cyclogenesis in that it removes all the simplifications and restrictive assumptions contained in previous formulations Linearized primitive equations in isentropic coordinates are integrated in time to find the most unstable eigenmode. The appropriate boundary condition for such a model applied either at z = 0 or z = h is derived. Results obtained in the β-plane with a realistically steep and high isolated topography are discussed in relation to the basic state characteristics. Limits and prospects with respect to theory verification are also discussed.
Abstract
Baroclinic instability in the presence of steep finite amplitude topography is studied in the primitive equation model. The quasi-geostrophic theory of Alpine cyclogenesis of Speranza et al. is reanalyzed and discussed in this context.
The present model is a generalization of the one used by Stone to include topographic effects, lateral shear of the basic wind, and/or lateral walls. We focus in particular on the differences between this formulation and the quasi-geostrophic one when the meridional scale of the topography is very small (of the order of 100 km). We find that only in the primitive equation model does a small-volume mountain, of height and width comparable with those of the Alps, introduce significant large-scale modifications to the baroclinic modes. The most unstable mode attains its maximum amplitude to the southern side of the mountain. We show that these results do not depend upon the specification of the lateral boundary conditions provided the basic state baroclinicity is meridionally confined.
Abstract
Baroclinic instability in the presence of steep finite amplitude topography is studied in the primitive equation model. The quasi-geostrophic theory of Alpine cyclogenesis of Speranza et al. is reanalyzed and discussed in this context.
The present model is a generalization of the one used by Stone to include topographic effects, lateral shear of the basic wind, and/or lateral walls. We focus in particular on the differences between this formulation and the quasi-geostrophic one when the meridional scale of the topography is very small (of the order of 100 km). We find that only in the primitive equation model does a small-volume mountain, of height and width comparable with those of the Alps, introduce significant large-scale modifications to the baroclinic modes. The most unstable mode attains its maximum amplitude to the southern side of the mountain. We show that these results do not depend upon the specification of the lateral boundary conditions provided the basic state baroclinicity is meridionally confined.
Abstract
Observational and numerical studies on Alpine cyclogenesis have shown that a developing baroclinic wave approaching the mountain region gives rise to a disturbance of dipolar structure, extending throughout the troposphere with horizontal scales comparable to the Rossby deformation radius. It is possible to interpret such disturbances as modifications of baroclinically unstable modes, induced by localized topography.
In the present approach, the effect of the mountain is introduced in a perturbative sense, in the framework of quasi-geostrophic theory. Even in this simple approach the spatial structure of the unstable modes is modified by a localized topography in the direction required in order to explain the observed features. In the case of a continuously stratified fluid, the basic characteristics of the observed vertical structure are also reproduced.
Abstract
Observational and numerical studies on Alpine cyclogenesis have shown that a developing baroclinic wave approaching the mountain region gives rise to a disturbance of dipolar structure, extending throughout the troposphere with horizontal scales comparable to the Rossby deformation radius. It is possible to interpret such disturbances as modifications of baroclinically unstable modes, induced by localized topography.
In the present approach, the effect of the mountain is introduced in a perturbative sense, in the framework of quasi-geostrophic theory. Even in this simple approach the spatial structure of the unstable modes is modified by a localized topography in the direction required in order to explain the observed features. In the case of a continuously stratified fluid, the basic characteristics of the observed vertical structure are also reproduced.
Abstract
The authors search the stationary solutions of the barotropic vorticity equation in spherical coordinates by numerically solving the equations with the Newton–Keller pseudoarclength continuation method. The solutions consist of planetary-scale Rossby waves superimposed on zonal wind profiles and forced by sinusoidal orography in near-resonance conditions. By varying the zonal wind strength across resonance, it is shown that multiple solutions with different wave amplitudes can be found: for small forcing and dissipation, the solution curve is the well-known bended resonance. The comparison between numerical results and theoretical predictions by a previously developed weakly nonlinear theory is successfully attempted.
The authors then extend the barotropic, weakly nonlinear theory to stationary Rossby waves forced by large-scale orography and dissipated by Ekman friction at the surface, in the framework of the quasigeostrophic model continuous in the vertical direction. The waves are superimposed on vertical profiles of zonal wind and stratification parameters taken from observations of the wintertime Northern Hemisphere circulation. In near-resonant conditions, the weakly nonlinear theory predicts multiple amplitude equilibration of the eddy field for a fixed vertical profile of the zonal wind. The authors discuss the energetics of the stationary waves and show that the form drag and Ekman dissipation can be made very small even if realistic values of the parameters are taken, at variance with the barotropic case.
This model is proposed as the theoretical base for such phenomena as atmospheric blocking, bimodality, and weather regimes.
Abstract
The authors search the stationary solutions of the barotropic vorticity equation in spherical coordinates by numerically solving the equations with the Newton–Keller pseudoarclength continuation method. The solutions consist of planetary-scale Rossby waves superimposed on zonal wind profiles and forced by sinusoidal orography in near-resonance conditions. By varying the zonal wind strength across resonance, it is shown that multiple solutions with different wave amplitudes can be found: for small forcing and dissipation, the solution curve is the well-known bended resonance. The comparison between numerical results and theoretical predictions by a previously developed weakly nonlinear theory is successfully attempted.
The authors then extend the barotropic, weakly nonlinear theory to stationary Rossby waves forced by large-scale orography and dissipated by Ekman friction at the surface, in the framework of the quasigeostrophic model continuous in the vertical direction. The waves are superimposed on vertical profiles of zonal wind and stratification parameters taken from observations of the wintertime Northern Hemisphere circulation. In near-resonant conditions, the weakly nonlinear theory predicts multiple amplitude equilibration of the eddy field for a fixed vertical profile of the zonal wind. The authors discuss the energetics of the stationary waves and show that the form drag and Ekman dissipation can be made very small even if realistic values of the parameters are taken, at variance with the barotropic case.
This model is proposed as the theoretical base for such phenomena as atmospheric blocking, bimodality, and weather regimes.
Abstract
In a preceding paper the authors showed that planetary waves of very different amplitudes can be sustained on the same configuration of the zonal wind by asymptotically balancing the energy contributions related to Ekman dissipation and orographic drag. The basic physical mechanism considered, namely, nonlinear self-interaction of the eddy field, was modeled in a vertically continuous quasigeostrophic model by means of a perturbative approach that relies on an ad hoc choice of the meridional profile of the wave field itself. Given the mathematical limitations of this approach, some important aspects of the mechanism of resonance bending were not explored; in particular, the sensitivity of stationary solutions to changes in the zonal wind profile, channel geometry, and physical parameters such as dissipation coefficients and mountain height.
In the present paper, the robustness of the mechanism of resonance folding by numerical means is analyzed, in the framework of both the barotropic and the two-level quasigeostrophic model. It is demonstrated that resonance bending is a generic property of the equations governing atmospheric motions on the planetary scale. In particular, it is shown that multiple stationary solutions can be achieved with realistic values of Ekman dissipation and mountain height in the context of the two-level quasigeostrophic model.
The authors formulate a weakly nonlinear theory that does not rely on any a priori assumptions about the meridional structure of the solution. Numerical and analytical results are compared, obtaining a satisfactory agreement in the parameter range in which the asymptotic theory is valid. The authors conclude that the present model is still a good candidate for the explanation of one of the most relevant statistical property of low-frequency variability at midlatitudes, namely, that large amplitude fluctuations of ultralong waves correspond to small variations of the zonal wind.
Abstract
In a preceding paper the authors showed that planetary waves of very different amplitudes can be sustained on the same configuration of the zonal wind by asymptotically balancing the energy contributions related to Ekman dissipation and orographic drag. The basic physical mechanism considered, namely, nonlinear self-interaction of the eddy field, was modeled in a vertically continuous quasigeostrophic model by means of a perturbative approach that relies on an ad hoc choice of the meridional profile of the wave field itself. Given the mathematical limitations of this approach, some important aspects of the mechanism of resonance bending were not explored; in particular, the sensitivity of stationary solutions to changes in the zonal wind profile, channel geometry, and physical parameters such as dissipation coefficients and mountain height.
In the present paper, the robustness of the mechanism of resonance folding by numerical means is analyzed, in the framework of both the barotropic and the two-level quasigeostrophic model. It is demonstrated that resonance bending is a generic property of the equations governing atmospheric motions on the planetary scale. In particular, it is shown that multiple stationary solutions can be achieved with realistic values of Ekman dissipation and mountain height in the context of the two-level quasigeostrophic model.
The authors formulate a weakly nonlinear theory that does not rely on any a priori assumptions about the meridional structure of the solution. Numerical and analytical results are compared, obtaining a satisfactory agreement in the parameter range in which the asymptotic theory is valid. The authors conclude that the present model is still a good candidate for the explanation of one of the most relevant statistical property of low-frequency variability at midlatitudes, namely, that large amplitude fluctuations of ultralong waves correspond to small variations of the zonal wind.
