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## 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.

## 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

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

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 dynamics of two-dimensional, shallow moist convection is examined with the use of a six-component spectral model. Latent heating effects are incorporated by assuming that upward motion is moist adiabatic and that downward motion is dry adiabatic. The resulting nondimensional system of equations has the same form as that for Bénard convection, with the moist effects included by replacing the Rayleigh number with a modified form.

The six-coefficient model contains a wide variety of multiple solutions with as many as 12 time-independent convective states and one conductive state occurring simultaneously. Temporally periodic solutions are also indicated, and some are found numerically that branch from stationary solutions at critical values of the external parameters. Only some of the solutions are linearly stable and hence observable, and we give a summary of the possible branching orders of these solutions. We find that the development of moist convection proceeds in the model via one of several available sequences of distinct transitions of the flow regime to increasingly complex structures.

## Abstract

The dynamics of two-dimensional, shallow moist convection is examined with the use of a six-component spectral model. Latent heating effects are incorporated by assuming that upward motion is moist adiabatic and that downward motion is dry adiabatic. The resulting nondimensional system of equations has the same form as that for Bénard convection, with the moist effects included by replacing the Rayleigh number with a modified form.

The six-coefficient model contains a wide variety of multiple solutions with as many as 12 time-independent convective states and one conductive state occurring simultaneously. Temporally periodic solutions are also indicated, and some are found numerically that branch from stationary solutions at critical values of the external parameters. Only some of the solutions are linearly stable and hence observable, and we give a summary of the possible branching orders of these solutions. We find that the development of moist convection proceeds in the model via one of several available sequences of distinct transitions of the flow regime to increasingly complex structures.

## Abstract

Cross sections of potential temperature, wind shell, and gradient Richardson number were constructed from data obtained during a Project HICAT flight and analyzed to determine the relationship to clear air turbulence in the stratosphere. CAT was found to be associated with strong baroclinic zones and with a critical value of the Richardson number of 0.25.

Energy budgets for five patches of turbulence associated with this outbreak of stratosphere clear air turbulence were also examined, and found to balance within 4–18% of the total rate of shear production.

## Abstract

Cross sections of potential temperature, wind shell, and gradient Richardson number were constructed from data obtained during a Project HICAT flight and analyzed to determine the relationship to clear air turbulence in the stratosphere. CAT was found to be associated with strong baroclinic zones and with a critical value of the Richardson number of 0.25.

Energy budgets for five patches of turbulence associated with this outbreak of stratosphere clear air turbulence were also examined, and found to balance within 4–18% of the total rate of shear production.

## 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.

## 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 set of conditions which justify the application of the Boussinesq approximation to compressible fluids is developed. Two cases are found and compared. In the first, in which the vertical scale of the motion can be of the same order of magnitude as the scale height of the medium, the perturbation momentum must be nondivergent and the effects of perturbations of pressure appear in several places. In the other case, where the vertical scale of the motion is much less than the scale height, the perturbation velocities are non-divergent and the perturbation pressure appears only in the pressure gradient force.

The approximate equations lead to linearized equations controlling the stability of wave motion which are formally equivalent to those for the same problem in the flow of a stratified medium which is incompressible in the sense that the flow is solenoidal. Thus, a variety of results about such motions are made applicable to the problems of convection and gravity wave motion in the atmosphere.

Various properties of the approximate equations are investigated; it is shown that acoustic modes are not permitted; quadratic forms which can serve as energies in various cases are developed; and integral methods of determining stability criteria are reviewed and applied.

In order to give the results wider applicability than to ideal gases, an ideal liquid is defined (*c _{p}* and the coefficients of expansion all being constant). The thermodynamic functions of this ideal liquid, including the entropy, internal energy and potential temperature, are determined explicitly.

## Abstract

A set of conditions which justify the application of the Boussinesq approximation to compressible fluids is developed. Two cases are found and compared. In the first, in which the vertical scale of the motion can be of the same order of magnitude as the scale height of the medium, the perturbation momentum must be nondivergent and the effects of perturbations of pressure appear in several places. In the other case, where the vertical scale of the motion is much less than the scale height, the perturbation velocities are non-divergent and the perturbation pressure appears only in the pressure gradient force.

The approximate equations lead to linearized equations controlling the stability of wave motion which are formally equivalent to those for the same problem in the flow of a stratified medium which is incompressible in the sense that the flow is solenoidal. Thus, a variety of results about such motions are made applicable to the problems of convection and gravity wave motion in the atmosphere.

Various properties of the approximate equations are investigated; it is shown that acoustic modes are not permitted; quadratic forms which can serve as energies in various cases are developed; and integral methods of determining stability criteria are reviewed and applied.

In order to give the results wider applicability than to ideal gases, an ideal liquid is defined (*c _{p}* and the coefficients of expansion all being constant). The thermodynamic functions of this ideal liquid, including the entropy, internal energy and potential temperature, are determined explicitly.

## Abstract

It has been suggested that the study of phase angles associated with the Fourier transforms of time series may yield information about the intermittent behavior of turbulent records. It is shown with numerical experiments that the phase angles and the fine-structure of the spectrum are both associated with the intermittency. The phase angles of turbulence appear to be nearly independent and uniformly distributed in the same sense that the spectrum has an approximate −5/13 power dependence on the wavenumber. But neither of these approximate models account for the observed intermittency. The intermittency therefore bears some relation to a higher order structure in Fourier space. The nature of this structure has not been found explicitly, although a qualitative explanation is offered.

## Abstract

It has been suggested that the study of phase angles associated with the Fourier transforms of time series may yield information about the intermittent behavior of turbulent records. It is shown with numerical experiments that the phase angles and the fine-structure of the spectrum are both associated with the intermittency. The phase angles of turbulence appear to be nearly independent and uniformly distributed in the same sense that the spectrum has an approximate −5/13 power dependence on the wavenumber. But neither of these approximate models account for the observed intermittency. The intermittency therefore bears some relation to a higher order structure in Fourier space. The nature of this structure has not been found explicitly, although a qualitative explanation is offered.