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James G. Vickroy and John A. Dutton

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.

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Donald H. Lenschow and John A. Dutton

Abstract

The problems and advantages of the bolometric method of measuring surface temperature from an airplane are discussed. It is shown that the airborne bolometer measures a weighted area-mean temperature which is a function of the surface temperature distribution and emissivity. For the employed wavelength of radiation (in the atmospheric “window”), the effect of atmospheric absorptivity is negligible under ordinary conditions for altitudes of 300 m or less, but can be an important consideration for flight levels above 300 m. From a series of flights over Southwestern Wisconsin it is concluded that the diurnal variation of surface temperature is greatest in flat farmland and sandy field areas, where a maximum difference of 29C occurred. Hilly woods and fields showed the smallest diurnal variation, with a maximum of 14C. The maximum standard deviation occurred in flat farmland area, with a value of 6C. A swampy area had a maximum value of 2C for a standard deviation.

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Richard D. Delay and John A. Dutton

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.

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John A. Dutton and George H. Fichtl

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 (cp 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.

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Jon M. Nese and John A. Dutton

Abstract

A dynamical systems approach is used to quantify the predictability of weather and climatic states of a low order, moist general circulation model. The effects on predictability of incorporating a simple oceanic circulation are evaluated. The predictability and structure of the model attractors are compared using Lyapunov exponents, local divergence rates, and the correlation and Lyapunov dimensions.

Lyapunov exponents quantify global, or time-averaged predictability, by measuring the mean rate of growth of small perturbations on an attractor, while local divergence rates quantify temporal variations of this error growth rate and thus measure local, or instantaneous, predictability.

Activating an oceanic circulation increases the average error doubling time of the atmosphere and the coupled ocean-atmosphere system by 10% while decreasing the variance of the largest local divergence rate by 20% . The correlation dimension of the attractor decreases slightly when an oceanic circulation is activated, while the Lyapunov dimension decreases more significantly because it depends directly on the Lyapunov exponents.

The average predictability of annually averaged states is improved by 25% when an oceanic circulation develops, and the variance of the largest local divergence rate also decreases by 25%. One-third of the yearly averaged states have local error doubling times larger than 2 years, indicating that annual averages may, at times, be predictable, even without predictable variations in external forcing. The dimensions of the attractors of the yearly averaged states are not significantly different than the dimensions of the attractors of the original model.

Arguably the most important contribution of this article is the demonstration that the local divergence rates provide a concise quantification of the variations of predictability on attractors and an efficient basis for comparing their local predictability characteristics. From a practical standpoint, local divergence rates might he computed to provide a real-time estimate of local predictability to accompany an operational forecast.

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Hampton N. Shirer and John A. Dutton

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.

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FRANK S. SECHRIST and JOHN A. DUTTON

Abstract

A new Lagrangian method of determining energy conversion rates for individual cyclones is developed and applied to the rapidly deepening storm of Nov. 29–30, 1963.

It is shown that 12-hr average conversion rates can be determined for arbitrary volumes of the atmosphere that move with the velocity of the mean wind. The method is straightforward, relatively easy to apply, and eliminates the necessity of evaluating boundary flux terms and other quantities that are difficult to measure accurately.

The new Lagrangian method has the advantage of providing vertical profiles of the conversion rates that exhibit revealing temporal changes. It appears that the storm initially acquired kinetic energy in the lowest layers southwest of the center. Later, the largest kinetic energy increases occurred south of the center at intermediate levels. By the time the storm reached maturity, the largest increases were taking place at high levels northeast of the storm while kinetic energy decreases occurred below.

Finally, 36-hr isentropic trajectories are used to trace parcels backward in time for the purpose of determining the source regions of air characterized by large values of kinetic energy. The results of this analysis indicate that a preexisting source of kinetic energy associated with the jet maximum northwest of the storm provided part of the storm's energy; the remainder was generated locally as ascending air parcels accelerated northeastward from the nearly barotropic region in the warm sector of the cyclone.

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John A. Dutton, Leonard J. Pietrafesa, and John T. Snow
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Jon M. Nese, Arthur J. Miller, and John A. Dutton

Abstract

A low-order moist general circulation model of the coupled ocean-atmosphere system is reexamined to determine the source of short-term predictability enhancement that occurs when an oceanic circulation is activated. The predictability enhancement is found to originate predominantly in thermodynamic processes involving changes in the mean hydrologic cycle of the model, which arise because the mean sea surface temperature is altered by the oceanic circulation. Thus, time-dependent sea surface temperature anomalies forced by anomalous geostrophic currents in the altered mean conditions do not contribute to the dominant ocean-atmosphere feed-back mechanism that causes the predictability enhancement in the model.

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Jon M. Nese, John A. Dutton, and Robert Wells

Abstract

Advancing knowledge about the phase space topologies of nonlinear hydrodynamic or dynamical systems has raised the question of whether the structure of the attractors in which the solutions are eventually confined can be characterized rigorously and economically. It is shown by applying the Lyapunov exponents, Lyapunov dimension, and correlation dimension to several low-order truncated spectral models that these quantities give useful information about the phase space structure and predictability characteristics of such attractors. The Lyapunov exponents measure the average exponential rate of convergence or divergence of nearby solution trajectories in an appropriate phase space. The Lyapunov dimension d L incorporates the dynamical information of the Lyapunov exponents to give an estimate of the dimension of the system attractor, while the correlation dimension v is a more geometrically motivated measure that is simple to compute and related to more classical dimensions.

The Lyapunov exponents detect bifurcations between solution regimes and also subtle predictability differences between attractors. As measures of chaotic attractor dimension, v>d L in all cases, and the ratio v/d L is smallest at values of the forcing just above the transition to chaos. Changes in the Lyapunov dimension are concentrated in a small range of forcing values, while the correlation dimension varies more uniformly. The value of d L is tied closely to the number of positive Lyapunov exponents, while v is more sensitive to the magnitude of the chaotic component of the system. Variations in these measures for a hierarchy of convection models support the idea that the appearance of strong chaos in two-dimensional models is truncation-related, and can be delayed to arbitrarily large forcing if enough modes are included.

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