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Edward N. Lorenz

Abstract

A rapid procedure for inverting a second-order finite-difference form of the two-dimensional del-square operator is presented. The procedure may be used whenever the precision of the computer significantly exceeds the required accuracy of the results; effectively it acquires its speed by using the otherwise unneeded power of the computer represented by the additional precision. A particular variant should prove especially useful in numerical weather prediction, in instances when storage space is at a premium.

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Edward N. Lorenz

Abstract

An approximate differential equation is presented, relating the change in speed of the zonal westerly winds to the contemporary zonal wind-speed and the meridional flow of absolute angular momentum. This equation is tested statistically by means of values of the momentum flow and the zonal wind-speed, computed with the aid of the geostrophic-wind approximation, from pressure and height data extracted from analyzed northern-hemisphere maps. The momentum flow is found to be positively correlated with the contemporary zonal wind-speed, and also with the contemporary change of the zonal wind-speed, in agreement with the approximate equation. The study suggests that the momentum flow may be a useful quantity for forecasting the zonal wind-speed. It also implies that an important part of the momentum flow is accomplished by means of large-scale horizontal eddies, whose forms are not obscured by the use of subjectively analyzed maps nor by the geostrophic-wind approximation.

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Edward N. Lorenz

Abstract

On the basis of five years of Northern Hemisphere isobaric height data, states of the atmosphere separated by 12 days or less are found on the average to resemble each other more closely than randomly selected states, even after adjustment for seasonal trend has been made. The existence of partial predictability of instantaneous weather patterns at least 12 days in advance is thereby confirmed.

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EDWARD N. LORENZ

Abstract

A time-differencing scheme consisting of an initializing step and N repetitions of a set of steps is proposed. For linear equations, the scheme is of Nth order. It is easily programmed and uses a minimal amount of storage space. The order may be changed by changing one parameter. An improved scheme is of Nth order even for nonlinear equations, for N≤4.

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Edward N. Lorenz

Abstract

No abstract available.

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Edward N. Lorenz

Abstract

We identify the slow manifold of a primitive-equation system with the set of all solutions that are completely devoid of gravity-wave activity. We construct a five-variable model describing coupled Rossby waves and gravity waves. Successive-approximation schemes designed to determine the slow manifold fail to converge when applied to the model, although they sometimes appear to converge before finally diverging. A noniterative scheme which demands only that the fast variables be functions of the slow variables yields a “Slowest invariant manifold,” which, however, is not unequivocally slow. We question whether the complete absence of gravity waves can be logically defined, and we note that the existence or nonexistence of a slow manifold does not depend upon the convergence or nonconvergence of a power series or a succession of approximations.

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Edward N. Lorenz

Abstract

Two studies that disagree as to whether a slow manifold is present in a particular low-order primitive equation model are compared. It is shown that the discrepancy occurs because of a difference of opinion as to what constitutes a slow manifold.

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Edward N. Lorenz

Abstract

Zonal flow resembling zonally averaged tropospheric motion in middle latitudes is usually barotropically stable, but zonal flow together with superposed neutral Rossby waves may be unstable with respect to further perturbations. Rossby's original solution of the barotropic vorticity equation is tested for stability, using beta-plane geometry. When the waves are sufficiently strong or the wavenumber is sufficiently high, the flow is found to be unstable, but if the flow is weak or the wavenumber is low, the beta effect may render the flow stable. The amplification rate of growing perturbations is comparable to the growth rate of errors deduced from large numerical models of the atmosphere. The Rossby wave motion together with amplifying perturbations possesses jet-like features not found in Rossby wave motion alone. It is suggested that barotropic instability is largely responsible for the unpredictability of the real atmosphere.

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Edward N. Lorenz

Abstract

Dynamical systems possessing regimes are identified with those where the state space possesses two or more regions such that transitions of the state from either region to the other are rare. Systems with regimes are compared to those where transitions are impossible.

A simple one-dimensional system where a variable is defined at N equally spaced points about a latitude circle, once thought not to possess regimes, is found to exhibit them when the external forcing F slightly exceeds its critical value F* for the appearance of chaos. Regimes are detected by examining extended time series of quantities such as total energy. A chain of k* fairly regular waves develops if F < F*, and F* is found to depend mainly upon the wavelength L* = N/k*, being greatest when L* is closest to a preferred length L 0. A display of time series demonstrates how the existence and general properties of the regimes depend upon L*.

The barotropic vorticity equation, when applied to an elongated rectangular region, exhibits regimes much like those occurring with the one-dimensional system. A first-order piecewise-linear difference equation produces time series closely resembling some produced by the differential equations, and it permits explicit calculation of the expected duration time in either regime. Speculations as to the prevalence of regimes in dynamical systems in general, and to the applicability of the findings to atmospheric problems, are offered.

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Edward N. Lorenz

Abstract

Variations of weather and climate are termed “forced” or “free” according to whether or not they are produced by variations in external conditions.

In many simple climate models, the poleward transport of sensible heat in the atmosphere has been treated as a diffusive process, and has been assumed to be proportional to the poleward temperature gradient. The validity of this assumption, for various space and time scales, is tested with 10 years of twice-daily upper level weather data. The space scales are defined by a spherical harmonic analysis, while the time scales are defined by a “poor man's spectral analysis.” The diffusive assumption is verified for the long-term average and the seasonal variations of the largest space scale, but it fails to hold for most of the remaining scales.

It is shown that diffusive behavior can be expected only for forced scales. It is suggested that most of the scales resolved by the data are free.

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