Search Results

You are looking at 1 - 10 of 22 items for

  • Author or Editor: Edward N. Lorenz x
  • All content x
Clear All Modify Search
Edward N. Lorenz

Abstract

Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.

A simple system representing cellular convection is solved numerically. All of the solutions are found to be unstable, and almost all of them are nonperiodic.

The feasibility of very-long-range weather prediction is examined in the light of these results.

Full access
Edward N. Lorenz

Abstract

A generalized vorticity equation for a two-dimensional spherical earth is obtained by eliminating pressure from the equations of horizontal motion including friction. The generalized vorticity equation is satisfied by formal infinite series representing the density and wind fields. The first few terms of a particular series solution are obtained explicitly. The series appear to converge near the north pole, and determine a model of a polar air mass. Within the air mass, the coldest winds are northeasterly and the warmest are southwesterly, while the coldest air of all is at the north pole. Heating occurs in the northwesterly winds and cooling in the southeasterlies, while aside from the effect of friction the air mass as a whole is cooled. The energy balance of the air mass is investigated. It is suggested that an analogous distribution of heating and cooling may be instrumental in maintaining the general circulation of the atmosphere.

Full access
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.

Full access
Edward N. Lorenz

Abstract

A two-layer quasi-geostrophic beta-plane model is converted into a moist general circulation model by including total water content as an additional prognostic variable. The water-vapor and liquid-water mixing ratios are determined diagnostically from the total-water mixing ratio and the saturation mixing ratio. The underlying surface is ocean, which exchanges water with the atmosphere through evaporation and precipitation. The circulation is driven by solar heating. Thermodynamic and radiative effects of water are included. The model is reduced to a low-order model by expressing each horizontal field in terms of seven orthogonal functions.

When horizontal variations of solar heating are suppressed, there are sometimes two stable steady states—a cold, rather cloudy state and a warm, nearly clear state. A cloud-albedo feedback process appears to be responsible for the multiple equilibria. With variable solar heating the model produces cyclones and anticyclones, with maxima of relative humidity and precipitation ahead of the cyclones, and minima ahead of the anticyclones.

Full access
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.

Full access
Edward N. Lorenz

Abstract

By means of truncated Fourier-Bessel series, a two-layer geostrophic “numerical-prediction” model with heating and friction is reduced to a set of eight ordinary nonlinear differential equations in eight dependent variables. These equations allow the presence of disturbances of a single wave number. They permit the occurrence of baroclinic but not barotropic instability. They possess appropriate energy invariants if heating and friction are temporarily suppressed.

The simplified equations are applied to the flow of a liquid in a symmetrically heated rotating basin. Exact solutions are determined for the steady Hadley and Rossby regimes, and the criterion for the stability of the Hadley regime is obtained. For high rotation rates the criterion for the disappearance of an established Rossby regime differs from the criterion for the onset of a Rossby regime.

The equations are modified to allow for the presence of several wave numbers simultaneously. Each wave number interacts with the zonal flow, but the interactions between wave numbers are omitted. The criteria for the transitions between wave numbers are then obtained.

The solutions agree qualitatively with Fultz's experiments in that with slow rotation there is no Rossby regime, with more rapid rotation the Rossby regime occurs with intermediate heating contrasts, and within the Rossby regime a smaller heating contrast leads to a higher wave number. It is concluded that the simplified equations are suitable for the study of baroclinic flow, and that the changes of regime are fundamental properties of the forced flow of a rotating fluid. It is suggested that the transitions in the experiments and the transitions described by the equations are manifestations of baroclinic instability having similar physical explanations.

Full access
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.

Full access
Edward N. Lorenz

Abstract

No abstract available.

Full access
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.

Full access
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.

Full access