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## Abstract

Properties of nonlinear quasi-geostrophic flow in unforced and in thermally-forced, dissipative modes are compared. The article is based on the philosophy that precise versions of the important problems of predictability and the theory of climate can be studied analytically with the quasi-geostrophic equation, regardless of whatever deficiencies it may have in representing atmospheric motion. The main result and contrast is that the trajectories of unforced flow are almost always recurrent in their spectral representation, returning infinitely often to a neighborhood of their initial points, and that all trajectories of forced, dissipative flow proceed eventually to the same limit set of measure zero in phase space.

The basic quasi-geostrophic model is extended in a number of ways. First, a global model is developed by making appropriate sign changes for the Southern Hemisphere, which produces a satisfactory spectral model even though discontinuities may appear at the equator. The efficacy of the model is illustrated with a generalization of the Rossby wave theory that gives a latitudinally variable wave speed depending on a latitudinally variable basic velocity.

Second, the boundary condition on the upper and lower surfaces is generalized considerably, so that temporally varying patterns of potential temperature perturbations and associated wind shears can be included at the boundaries. The changes at the boundaries are internally controlled or forced by the imposed heating field.

The statistical properties of the usual unforced quasi-geostrophic flow are considered in phase space, and it is shown that the Poincaré recurrence theorem applies and that long-term averages along the trajectories exist even though the flows are not ergodic.

The thermally-forced model is developed by adding a Newtonian heating term to the First Law and by adding a dissipative term to the vorticity equation. In this model every initial set is mapped into a set of vanishing measure as *t* → ∞. Moreover, it is shown that all trajectories are eventually trapped in a region of phase space specified by the rates of heating and dissipation. The limit properties of the trajectories are examined and it is shown that each has at least one limit point, so that all trajectories not asymptotic to a stationary point are repetitive. However, it is also shown that cycles can occur only in the limit set of measure zero.

Some errors in Part I are corrected in an Appendix.

## Abstract

Properties of nonlinear quasi-geostrophic flow in unforced and in thermally-forced, dissipative modes are compared. The article is based on the philosophy that precise versions of the important problems of predictability and the theory of climate can be studied analytically with the quasi-geostrophic equation, regardless of whatever deficiencies it may have in representing atmospheric motion. The main result and contrast is that the trajectories of unforced flow are almost always recurrent in their spectral representation, returning infinitely often to a neighborhood of their initial points, and that all trajectories of forced, dissipative flow proceed eventually to the same limit set of measure zero in phase space.

The basic quasi-geostrophic model is extended in a number of ways. First, a global model is developed by making appropriate sign changes for the Southern Hemisphere, which produces a satisfactory spectral model even though discontinuities may appear at the equator. The efficacy of the model is illustrated with a generalization of the Rossby wave theory that gives a latitudinally variable wave speed depending on a latitudinally variable basic velocity.

Second, the boundary condition on the upper and lower surfaces is generalized considerably, so that temporally varying patterns of potential temperature perturbations and associated wind shears can be included at the boundaries. The changes at the boundaries are internally controlled or forced by the imposed heating field.

The statistical properties of the usual unforced quasi-geostrophic flow are considered in phase space, and it is shown that the Poincaré recurrence theorem applies and that long-term averages along the trajectories exist even though the flows are not ergodic.

The thermally-forced model is developed by adding a Newtonian heating term to the First Law and by adding a dissipative term to the vorticity equation. In this model every initial set is mapped into a set of vanishing measure as *t* → ∞. Moreover, it is shown that all trajectories are eventually trapped in a region of phase space specified by the rates of heating and dissipation. The limit properties of the trajectories are examined and it is shown that each has at least one limit point, so that all trajectories not asymptotic to a stationary point are repetitive. However, it is also shown that cycles can occur only in the limit set of measure zero.

Some errors in Part I are corrected in an Appendix.

## Abstract

Aperiodic solutions to spectrally truncated models based on the vorticity equation are considered for the case of a zonal flow interacting nonlinearly with two other components both having the same zonal wavenumber. It is shown that all such aperiodic trajectories proceed asymptotically to either a stationary point in the phase space of coefficients or to a periodic solution with steady amplitudes.

It is also shown that the set of such solutions is of measure zero on surfaces of constant energy in phase space. Thus if the initial coefficients for a nonlinear, three-component flow are selected at random, then the resulting flow will in all probability be periodic.

## Abstract

Aperiodic solutions to spectrally truncated models based on the vorticity equation are considered for the case of a zonal flow interacting nonlinearly with two other components both having the same zonal wavenumber. It is shown that all such aperiodic trajectories proceed asymptotically to either a stationary point in the phase space of coefficients or to a periodic solution with steady amplitudes.

It is also shown that the set of such solutions is of measure zero on surfaces of constant energy in phase space. Thus if the initial coefficients for a nonlinear, three-component flow are selected at random, then the resulting flow will in all probability be periodic.

The CAT in the sky, which has created a variety of problems since aviation began, is being attacked by an extensive measurement program sponsored by the United States Air Force under the code name ALLCAT.

Clear air turbulence data gathering programs at all altitudes important to modern aviation are described, and some of the meteorological applications of the data are pointed out. It is shown how a properly instrumented airplane can be used to obtain actual gust velocities. The power spectral and exceedance statistics methods used to incorporate gust measurement data in aircraft design procedure are summarized and illustrated.

The article concludes that the program will be an important step in taming all the CATS.

The CAT in the sky, which has created a variety of problems since aviation began, is being attacked by an extensive measurement program sponsored by the United States Air Force under the code name ALLCAT.

Clear air turbulence data gathering programs at all altitudes important to modern aviation are described, and some of the meteorological applications of the data are pointed out. It is shown how a properly instrumented airplane can be used to obtain actual gust velocities. The power spectral and exceedance statistics methods used to incorporate gust measurement data in aircraft design procedure are summarized and illustrated.

The article concludes that the program will be an important step in taming all the CATS.

## Abstract

The quasi-geostrophic theory leads to a single nonlinear partial differential equation for a streamfunction giving geostrophic velocity fields presumed to resemble the synoptic scales of atmospheric motion. This article is concerned with demonstrating that the quasi-geostrophic problem is well-posed mathematically, in the sense that solutions exist, and that they are continuously dependent on the initial data. The model studied is comprised of the quasi-geostrophic equation subject to the severe boundary condition that an isentrope coincides with the earth's surface.

The main technique is the use of the eigenfunctions of an elliptic operator appearing within the quasi-geostrophic equation. These eigenfunctions provide the basis for a spectral model, which can be truncated to include a finite number of scales. The convergence properties of the solutions to the truncated model allow the existence of solutions to the entire model to be inferred with the methods of functional analysis. Thus, the conclusions reached are relative to generalized solutions and to the usual norm in the Hilbert space of quadratically integrable functions.

The results have applications to the study of atmospheric predictability.

## Abstract

The quasi-geostrophic theory leads to a single nonlinear partial differential equation for a streamfunction giving geostrophic velocity fields presumed to resemble the synoptic scales of atmospheric motion. This article is concerned with demonstrating that the quasi-geostrophic problem is well-posed mathematically, in the sense that solutions exist, and that they are continuously dependent on the initial data. The model studied is comprised of the quasi-geostrophic equation subject to the severe boundary condition that an isentrope coincides with the earth's surface.

The main technique is the use of the eigenfunctions of an elliptic operator appearing within the quasi-geostrophic equation. These eigenfunctions provide the basis for a spectral model, which can be truncated to include a finite number of scales. The convergence properties of the solutions to the truncated model allow the existence of solutions to the entire model to be inferred with the methods of functional analysis. Thus, the conclusions reached are relative to generalized solutions and to the usual norm in the Hilbert space of quadratically integrable functions.

The results have applications to the study of atmospheric predictability.

## Abstract

An analytical model of the globally averaged surface temperature response to changes in radiative forcing induced by greenhouse gases is developed from a time-dependent version of the global energy budget. The model clarifies the role of feedback and system heat capacity in controlling the magnitude and rate of response.

Observed seasonal changes in surface temperature, radiative fluxes, and planetary albedo are combined to estimate the atmospheric feedback and the net gain of the system. A simple model of ocean upwelling and diffusion then yields an estimate of the heat capacity and thus the time constant of the system. The observed global temperature change from 1900 to 1990 is used to calibrate the model and refine the estimate of the time constant. The model provides a framework for comparing numerical models, including time-dependent ocean-atmosphere models used by the Intergovernmental Panel on Climate Change to estimate expected global temperature changes.

When integrable analytical approximations to greenhouse-forcing scenarios are combined with the model, completely analytical representations of the global temperature change are readily obtained, yielding numerical estimates that agree with those of more complex models.

## Abstract

An analytical model of the globally averaged surface temperature response to changes in radiative forcing induced by greenhouse gases is developed from a time-dependent version of the global energy budget. The model clarifies the role of feedback and system heat capacity in controlling the magnitude and rate of response.

Observed seasonal changes in surface temperature, radiative fluxes, and planetary albedo are combined to estimate the atmospheric feedback and the net gain of the system. A simple model of ocean upwelling and diffusion then yields an estimate of the heat capacity and thus the time constant of the system. The observed global temperature change from 1900 to 1990 is used to calibrate the model and refine the estimate of the time constant. The model provides a framework for comparing numerical models, including time-dependent ocean-atmosphere models used by the Intergovernmental Panel on Climate Change to estimate expected global temperature changes.

When integrable analytical approximations to greenhouse-forcing scenarios are combined with the model, completely analytical representations of the global temperature change are readily obtained, yielding numerical estimates that agree with those of more complex models.

The atmospheric sciences, along with other disciplines, are today contemplating strong competition for resources, exciting scientific challenges, and the need to assess opportunities and priorities. This review is intended to stimulate thought, discussion, and the active participation of the community in charting a course into the next century.

Current efforts concentrate on mesoscale phenomena and severe weather, climate dynamics and prediction, the interplay of chemical and physical processes, the interactions of planetary atmospheres with ionospheres and solar processes, and the predictability of chaotic systems. These efforts are the foundation for initiatives involving global and regional climate change, mesoscale research and prediction, and the modernization of the National Weather Service.

Contemporary imperatives include developing leadership and management capabilities, making the requisite investments in advanced data and information systems, enabling the prediction of predictability, and developing appropriate approaches to education and to cooperation with other disciplines. Most of all, the atmospheric sciences community must learn to be more effective in determining priorities and in developing compelling rationales for obtaining the resources necessary to pursue signal opportunities for scientific advance and service to the nation.

The atmospheric sciences, along with other disciplines, are today contemplating strong competition for resources, exciting scientific challenges, and the need to assess opportunities and priorities. This review is intended to stimulate thought, discussion, and the active participation of the community in charting a course into the next century.

Current efforts concentrate on mesoscale phenomena and severe weather, climate dynamics and prediction, the interplay of chemical and physical processes, the interactions of planetary atmospheres with ionospheres and solar processes, and the predictability of chaotic systems. These efforts are the foundation for initiatives involving global and regional climate change, mesoscale research and prediction, and the modernization of the National Weather Service.

Contemporary imperatives include developing leadership and management capabilities, making the requisite investments in advanced data and information systems, enabling the prediction of predictability, and developing appropriate approaches to education and to cooperation with other disciplines. Most of all, the atmospheric sciences community must learn to be more effective in determining priorities and in developing compelling rationales for obtaining the resources necessary to pursue signal opportunities for scientific advance and service to the nation.

## Abstract

The variance spectrum of velocities in a non-homogeneous, compressible fluid does not represent the wave-number distribution of kinetic energy, as it does in incompressible, homogeneous (constant density) fluids. Use of a truncated Fourier transform and the assumption that the flow occurs in a finite area show that the kinetic energy spectrum in the former case is the co-spectrum between the velocity and the momentum. The Navier-Stokes equations are used to study the time rates of change of the kinetic energy spectrum produced by the various physical effects contained in those equations. Introduction of the assumption of homogeneity and incompressibility in the equations derived here gives the same qualitative results as Batchelor's (1953) study of the time rate of change of the spectrum of turbulent flow. Kinetic energy in a compressible, non-homogeneous fluid can draw on internal and potential energy, but these energy sources are not available to flow in incompressible, homogeneous fluids. It is shown that compressibility effects are not important in the action of the inertial or viscous effects on the total kinetic energy.

## Abstract

The variance spectrum of velocities in a non-homogeneous, compressible fluid does not represent the wave-number distribution of kinetic energy, as it does in incompressible, homogeneous (constant density) fluids. Use of a truncated Fourier transform and the assumption that the flow occurs in a finite area show that the kinetic energy spectrum in the former case is the co-spectrum between the velocity and the momentum. The Navier-Stokes equations are used to study the time rates of change of the kinetic energy spectrum produced by the various physical effects contained in those equations. Introduction of the assumption of homogeneity and incompressibility in the equations derived here gives the same qualitative results as Batchelor's (1953) study of the time rate of change of the spectrum of turbulent flow. Kinetic energy in a compressible, non-homogeneous fluid can draw on internal and potential energy, but these energy sources are not available to flow in incompressible, homogeneous fluids. It is shown that compressibility effects are not important in the action of the inertial or viscous effects on the total kinetic energy.

## Abstract

The steady solutions and their stability properties are investigated for a low-order spectral model of a forced, dissipative, nonlinear, quasi-geostrophic flow. A zonal flow is modified by two smaller scale disturbances in the model.

If only the zonal component (or only the smallest scale component) is forced, then the stationary solution is unique, always locally stable, and globally stable for weak forcing.

There is also a unique locally stable stationary solution for weak forcing of only the middle component. But as this forcing exceeds a critical value, a supercritical bifurcation to new solutions appears.

The entire solution surface for forcing of the zonal and middle components can be displayed graphically and is a form of the well-known cusp catastrophe surface. For forcing of all three components, the morphogenesis set is more complex, containing regions in which there are one, three or five solutions.

Numerical integrations of the phase-sparce trajectories of the solutions reveal that for forcing of the zonal and middle components 1) domains of attraction of stable steady solutions contain neighborhoods near the unstable steady solution, 2) there is a region near the cusp in which initial points produce periodic solutions, and 3) initial points further away from the cusp yield trajectories that quickly approach stable steady solutions.

The conclusion is that any successful theory of atmospheric climate will have to contend with multiple solutions and changing domains of attraction as external parameters are varied.

## Abstract

The steady solutions and their stability properties are investigated for a low-order spectral model of a forced, dissipative, nonlinear, quasi-geostrophic flow. A zonal flow is modified by two smaller scale disturbances in the model.

If only the zonal component (or only the smallest scale component) is forced, then the stationary solution is unique, always locally stable, and globally stable for weak forcing.

There is also a unique locally stable stationary solution for weak forcing of only the middle component. But as this forcing exceeds a critical value, a supercritical bifurcation to new solutions appears.

The entire solution surface for forcing of the zonal and middle components can be displayed graphically and is a form of the well-known cusp catastrophe surface. For forcing of all three components, the morphogenesis set is more complex, containing regions in which there are one, three or five solutions.

Numerical integrations of the phase-sparce trajectories of the solutions reveal that for forcing of the zonal and middle components 1) domains of attraction of stable steady solutions contain neighborhoods near the unstable steady solution, 2) there is a region near the cusp in which initial points produce periodic solutions, and 3) initial points further away from the cusp yield trajectories that quickly approach stable steady solutions.

The conclusion is that any successful theory of atmospheric climate will have to contend with multiple solutions and changing domains of attraction as external parameters are varied.

## Abstract

A truncated spectral model of the forced, dissipative, barotropic vorticity equation on a cyclic β-plane is examined for multiple stationary and periodic solutions. External forcing on one scale of the motion provides a barotropic analog to thermal heating.

For forcing of any (finite) magnitude at the maximum or minimum scale in the truncation, the truncated solution converges in the limit as *t* → ∞ to the known solution of the corresponding linear model. If the forcing is constant, this limit solution represents a globally attracting stationary point in phase space. These results extend the well-known spectral blocking theorem of Fjortøft (1953) to forced, dissipative flows.

The main results, however, obtain from a low-order model describing two disturbance components interacting with a constant, forced, basic-flow component of intermediate scale. The zonal dependence of either the basic flow or the disturbances is flexible and determined by the choice of component wave vectors. For low-wavenumber disturbances and β ≠ 0, the basic flow represents a unique stationary solution, which becomes unstable when the forcing exceeds a critical value. An application of the Hopf bifurcation theorem in the neighborhood of critical forcing reveals the existence of a periodic solution or limit cycle, which is then derived explicitly in phase space as a closed circular orbit whose frequency is described by a linear combination of the normal-mode Rossby-wave frequencies.

The limit cycle radius, which physically represents the ultimate enstrophy of the disturbances, can be depicted as a response surface on the control plane defined by the independent forcing and beta parameters. If the forcing is zonally dependent, the response surface may exhibit a pronounced fold, which arises from the existence of a snap-through bifurcation. The projection of this fold onto the parameter control plane defines a bimodal or hysteresis region in which multiple stable solutions exist for given parameters. The boundary of the hysteresis region represents parameter states at which the model can exhibit sudden flow regime transitions, analogous to those observed in the laboratory rotating annulus.

This study demonstrates that the degree of nonlinearity, the scale of the forcing, and the spatial dependence of the disturbances and the forcing all crucially influence both the multiplicity and temporal nature of the stable limit solutions in a low-order, forced, dissipative model. Thus, choices in this rather complex array of physical degrees of freedom must be carefully considered in any model of the long-term evolution of large-scale atmospheric flow.

## Abstract

A truncated spectral model of the forced, dissipative, barotropic vorticity equation on a cyclic β-plane is examined for multiple stationary and periodic solutions. External forcing on one scale of the motion provides a barotropic analog to thermal heating.

For forcing of any (finite) magnitude at the maximum or minimum scale in the truncation, the truncated solution converges in the limit as *t* → ∞ to the known solution of the corresponding linear model. If the forcing is constant, this limit solution represents a globally attracting stationary point in phase space. These results extend the well-known spectral blocking theorem of Fjortøft (1953) to forced, dissipative flows.

The main results, however, obtain from a low-order model describing two disturbance components interacting with a constant, forced, basic-flow component of intermediate scale. The zonal dependence of either the basic flow or the disturbances is flexible and determined by the choice of component wave vectors. For low-wavenumber disturbances and β ≠ 0, the basic flow represents a unique stationary solution, which becomes unstable when the forcing exceeds a critical value. An application of the Hopf bifurcation theorem in the neighborhood of critical forcing reveals the existence of a periodic solution or limit cycle, which is then derived explicitly in phase space as a closed circular orbit whose frequency is described by a linear combination of the normal-mode Rossby-wave frequencies.

The limit cycle radius, which physically represents the ultimate enstrophy of the disturbances, can be depicted as a response surface on the control plane defined by the independent forcing and beta parameters. If the forcing is zonally dependent, the response surface may exhibit a pronounced fold, which arises from the existence of a snap-through bifurcation. The projection of this fold onto the parameter control plane defines a bimodal or hysteresis region in which multiple stable solutions exist for given parameters. The boundary of the hysteresis region represents parameter states at which the model can exhibit sudden flow regime transitions, analogous to those observed in the laboratory rotating annulus.

This study demonstrates that the degree of nonlinearity, the scale of the forcing, and the spatial dependence of the disturbances and the forcing all crucially influence both the multiplicity and temporal nature of the stable limit solutions in a low-order, forced, dissipative model. Thus, choices in this rather complex array of physical degrees of freedom must be carefully considered in any model of the long-term evolution of large-scale atmospheric flow.